16th November 2006 9:21 WSPC - Proceedings Trim Size: 9in x 6in BeyrouthWSf

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COMPLEX CURVE SINGULARITIES: A BIASED INTRODUCTION

Bernard TEISSIER

Institut mathématique de Jussieu, UMR 7586 du C.N.R.S., Case 7012, 2 Place Jussieu, 75251 Paris Cedex 05, France E-mail: teissier@math.jussieu.fr

The goal of this text is to provide an introduction to the local study of singular curves in complex analytic geometry. It contains resolution of singularities, the Newton polygon and the Newton parametrization, the classical Newton-Puiseux invariants, the semigroup associated to a branch as well as the specialization to the corresponding monomial curve, and a proof of Bézout’s theorem. It ends with the computation of the class of a projective plane curve. Some familiarity with the basic concepts of commutative algebra and complex analytic geometry is assumed.

Keywords: curve singularities, Newton-Puiseux, monomial curves, Bézout Theorem

1. What is a curve?

In these lectures^{a} I will discuss singular points of complex analytic curves. A complex curve may locally be regarded as a family of points in complex affine space A^{d}(C) depending algebraically or analytically on one complex parameter.

• The dependence may be explicit, which means that the coordinates of our points depend explicitely on one parameter, as in:

\begin{aligned}z_{1} & = z_{1}(t) \\ z_{2} & = z_{2}(t) \\ ⋮ & = ⋮ \\ z_{d} & = z_{d}(t) \end{aligned}

aThis text is an expansion of the notes of a course given at the CIMPA-LEBANON school in Beyrouth in July 2004 and lectures at the ICTP College on Singularities.

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where the z_{i}(t) may be any functions C → C although here, since we said we would consider families of points depending analytically on the parameter, we shall consider only polynomials or convergent power series. In this case the functions z_{i}(t) may be defined only in a neighborhood of some point, which we will usually assume to be the origin t = 0 and it is convenient to assume that z_{i}(0) = 0 for all i; one may reduce to this case by a translation on the coordinates z_{i} and t. The curves which appear naturally are usually finite unions of parametrized curves as above. The parametrized curves are then the irreducible components or branches of the union. A curve is non singular at the origin if and only if it has only one component, for which the minimum of the t-adic orders of the z_{i}(t) is equal to one. By the implicit function theorem, this means that the curve is locally analytically isomorphic to the complex line.

A germ of curve at a point (which we take to be the origin) is an equivalence class of curves given parametrically or by equations in an open neighborhood of the origin. Two such objects defined respectively on U and U′ are equivalent if their restrictions to a third neighborhood U′′ ⊂ U ∩ U′ of the origin coincide. Of course when we talk of germs we think of representatives in some ”sufficiently small” neighborhood of 0. Because of analyticity, to give a germ is equivalent to giving the convergent power series parametrizing the branches of the curve at the origin in some coordinate system.

• A curve may also be given implicitely, which means that it is given by equations.

Here the fact that we are dealing with a curve should manifest itself in the fact that we have d − 1 equations in d variables, so that implicitely all the variables depend on one of them. However, the situation is not so simple in general, and a curve may need more than d − 1 equations.

It is a fundamental fact of the theory of analytic curves that each germ can be decomposed uniquely as a union of irreducible germs, and that on an irreducible germ all coordinates can be expressed as convergent power series in the n-th root of one of them, for some integer m. This is often called the Newton-Puiseux theorem. The simplest case is that of a plane algebraic curve in the 2-dimensional affine space A^{2}(C), defined by an equation f(x, y) = 0, where f ∈ C[x, y] is a polynomial:

f(x, y) = a_{0}(x)y^{n} + a_{1}(x)y^{n−1} + ⋯ + a_{n}(x),

with a_{i}(x) ∈ C[x], a_{0}(x) ≠ 0. Recall that C[z_{1}, …, z_{d}], C{z_{1}, …, z_{d}} and

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C[[z_{1}, …, z_{d}]] denote respectively the ring of polynomials in d variables with complex coefficients, the ring of convergent complex power series in d variables and the ring of formal power series in d variables with complex coefficients. The degree n in y of the polynomial is the number of solutions in y (counted with multiplicities) for any fixed value x_{0} of x which is “sufficiently general” in the sense that a_{0}(x_{0}) ≠ 0. The curve is non-singular at the origin if at least one of the derivatives \frac{∂f}{∂x}, \frac{∂f}{∂y} does not vanish at the origin. All this remains valid locally if f(x, y) is a convergent power series in x, y such that f(0, 0) = 0, thanks to the Weierstrass preparation theorem which asserts that, if f(0, y) is not identically zero, which we can always assume at the cost of a linear change of variables, then f(x, y) can be written

f(x, y) = u(x, y)(y^{n} + a_{1}(x)y^{n−1} + ⋯ + a_{n}(x)),

with a_{i}(x) ∈ C{x} where u(x, y) is a convergent series which does not vanish in a neighborhood of the origin. This means that u(x, y) is an invertible element in the ring C{x, y} and also that the curve defined by f(x, y) = 0 is just as well defined by the polynomial in y, in a sufficiently small neighborhood of the origin.

We shall see later an interpretation for the total degree max(i + j) for monomials x^{i}y^{j} appearing in a polynomial f(x, y)). To a parametrized plane curve x(t), y(t), one can associate a series f(x, y) such that f(x(t), y(t)) ≡ 0, by a process of elimination, to which we shall come back later. If the exponents of t appearing in the two series have no common divisor, this series is an irreducible element in the ring C{x, y}; it is not a product of two other series vanishing at the origin. Conversely the set of zeroes f(x, y) = 0 of a convergent power series in two variables defines coincides with a finite number of parametrized analytic curves in a small neighborhood of the origin of the plane, (the branches of the curve defined by the equation). The construction will be detailed below. The number r of these branches is given by the decomposition of f(x, y) into a product f_{1}^{a_{1}} … f_{r}^{a_{r}} of powers of irreducible elements in the ring C{x, y}.

We shall deal here with reduced curves, which means that all the a_{i} are equal to one or, as we shall see that the exponents appearing in the power series x(t), y(t) of a parametrization of each branch, taken all together, have no non trivial common divisor.

The description of a curve in A^{d}(C) for d ≥ 3 is much more complicated in general. One can prove that it requires at least d − 1 equations, but it may require much more, and then what is important is the ideal I which

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these equations generate in the ring C[z_{1}, …, z_{d}] or the ring C{z_{1}, …, z_{d}}. In fact, it is the quotient ring C[z_{1}, …, z_{d}]/I, or C{z_{1}, …, z_{d}}/I which we consider, and the fact that we a re dealing with reduced complex curves is translated into the fact that we suppose that it is a reduced C-algebra with Krull dimension 1. In the case of plane curves, this ideal is a principal ideal so we have one equation only; more precisely, in the case of polynomials, this equation is unique up to multiplication by a nonvanishing polynomial, i.e., a nonzero constant. In fact, it is the quotient ring C[z_{1}, …, z_{d}]/I, or C{z_{1}, …, z_{d}}/I which we consider.

Assuming that we are in the analytic (as opposed to algebraic) case and the ring O = C{z_{1}, …, z_{d}}/I is an integral domain of dimension one, we can bring together the equational and parametric representations together in the following diagram of C-algebras:

C{z_{1}, …, z_{d}} → O ⊂ C{t},

where the first map is the surjection with kernel I describing O as a quotient, and the second one is determined by: z_{i} ↦ z_{i}(t) 1 ≤ i ≤ d.

The conclusion is that a complex curve, locally, should be thought of as a one-dimensional reduced local C-algebra O, which is the localization of a C-algebra of finite type (algebraic case), or is a quotient of a convergent power series ring C{z_{1}, …, z_{d}} (analytic case) or of a formal power series ring C[[z_{1}, …, z_{d}]] (formal, or algebroid, case).

Then we can consider it as given by an ideal in a regular local ring, namely the kernel of the map defining it as a quotient of a localization of a polynomial ring, or as a subring of a regular one-dimensional semi-local ring (its normalization) \overline{O}, which in the analytic or formal case corresponds to the parametrization. It is a theorem that in the cases considered here, the normalization \overline{O} is a finitely generated O-module.

2. What does one do with curves?

Real curves (often analytic) appear everywhere in mechanics as trajectories, and complex curves appear everywhere in Mathematics as soon as points depend on one parameter; for example given a family of square complex matrices depending on a complex parameter, the family of the eigenvalues lies on a complex curve. In order to study a real analytic curve, it is often useful to look at its complexification.

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If a complex algebraic group G acts algebraically on a variety X, one may study the action by restricting it to one-parameter subgroups C^{∗} ⊂ G and the the orbit of each point x ∈ X is a curve. Its closure in X is in general singular.

More generally, the closure of a non singular curve will in general have singularities.

We have seen also non singular algebraic curves which are themselves algebraic groups; elliptic curves. However, if we want to understand the totality of elliptic curves, we must also consider their singular limits.

Curves also appear naturally in inductive steps in algebraic geometry: non-singular surfaces are studied in large part through the families of curves which they contain, and singular curves must appear in these families.

To study the geometry of a non singular surface S, it is natural to project it to a non-singular space of the same dimension. The set of points of S where the projection p: S → P^{2} is not a local isomorphisme, the critical locus of the projection, is a curve (if it is not empty) which contains much information about the geometry of S. The critical locus may be a non singular curve, but its image under projection p will in general be singular.

More generally, even if one is interested in non singular curves only, their plane projections will in general be singular, having at least ordinary double points or nodes

In these lectures, I will mostly study different ways to transform curves into other curves, by deformation either of the parametric representation or of the equations, by taking the transforms of curves in A_{k}^{d} under maps Z → A_{k}^{d}, and by projection (in the case of non plane curves).

Since the passage from the implicit presentation to the parametrization uses some form of the implicit function theorem, I will mostly work in the context of complex analytic functions, for which there is an implicit function theorem which does not exist for polynomials. If one wishes to work over a field different from C, one could replace convergent power series with formal power series with coefficients in a field, keeping most of the algebra but losing a lot of geometry or one could work in the henselizations of the polynomial ring and its quotients, keeping just about everything. If the field has positive characteristic new phenomena appear: Newton-Puiseux’s theorem is no longer true (see section ?? below).

Let us call analytic algebra any C-algebra which is a quotient of a convergent power series ring C{z_{1}, …, z_{d}}. To any localization R_{m} of a finitely generated local algebra R over the field of complex numbers at one of its

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maximal ideals is associated in a unique way (up to unique isomorphism) an analytic algebra R_{m}^{h}, which has the property that any C-algebra morphism R_{m} → A from R_{m} to an analytic algebra A factors in a unique manner R_{m} → R_{m}^{h} → A where R_{m}^{h} → A is a morphism of analytic algebras.

The candidate for R^{h} is simple to see: write a presentation R = C[T_{1}, …, T_{n}]/I, where I is generated by finitely many polynomials. The maximal ideal m of R is the image of a maximal ideal m̃ of C[T_{1}, …, T_{d}]. By the nullstellensatz, the ideal m̃ corresponds to a point (a_{1}, …, a_{d}) of the affine space A^{d}(C). Set R^{h} = C{T_{1} − a_{1}, …, T_{d} − a_{d}}/I^{h}, where I^{h} is the ideal generated in C{T_{1} − a_{1}, …, T_{d} − a_{d}} by the polynomials which generate I.

In the case of curves, the first serious difficulty comes from the fact that an irreducible polynomial P (x, y) ∈ C[x, y] may well become reducible in C{x, y} = C[x, y]_{(x,y)}^{h}. In other words, the analytization of a local integral C-algebra may not be integral. Consider for example the nodal cubic with equation

y^{2} − x^{2} − x^{3} = 0;

it is an irreducible affine plane curve, but the image of its equation in C{x, y} by the natural injection C[x, y] ⊂ C{x, y} factors as

y^{2} − x^{2} − x^{3} = (y + x\sqrt{1 + x})(y − x\sqrt{1 + x}).

The interaction between the global invariants of a plane projective curve and its singularities is also an important theme requiring the local study of singularities:

- We know how to compute the class and the genus of a non singular plane projective curve of degree d. Assume now that it is singular; how does it affect the formulas for the class and genus?

- The single most important result about plane curves is Bézout’s theorem, which is the generalization of the fundamental theorem of algebra: Given two plane projective curves C and C′ of degrees d and d′ having no common component, the number of their points of intersection counted with multiplicity is the product dd′.

At the points of intersection where both curves are not regular and meeting transversally, how does one properly count the intersection multiplicity? can one effectively count it, given the equations of the two curves? In fact, as we shall see, the best situation to compute the intersection multiplicity is to have one curve given parametrically and the other given implicitely, although if both are given parametrically, there is a formula, due to Max

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Noether (see [M]). What is the geometric meaning of intersection multiplicity?

We know that we can normalize an algebraic curve to obtain a non singular algebraic curve, and it can be shown that the same is true in complex analytic geometry. The parametrization of an analytic curve mentioned above is in fact its normalization, provided we take care that the powers of t appearing in the series are coprime; if this is the case, the inclusion O ⊂ C{t} induces an isomorphism of fraction fields and is therefore the normalization of O. The normalization, however, is a priori difficult to compute from the equations of our curve.

An algorithm to do precisely this, in fact to compute a parametrization from an equation of a plane curve, was given by Newton; it is based on the Newton polygon.

3. Newton’s study of plane curve singularities

Let us recall that the ring of power series in fractional powers with fixed denominator m of a variable x with fixed denominator m is defined as C[[x^{\frac{1}{m}}]] = C[[x]][T]/(T^{m} − x), and similarly for C\{x^{\frac{1}{m}} \}. The notation x^{\frac{1}{m}} describes a multi-valued function of x, defined as a function not on C but on an m-fold ramified covering of C. The various ”determinations” are exchanged by multiplying one of them by an m-th root of unity. The work of Newton and Puiseux shows that functions on a branch can be viewed as functions of x^{\frac{1}{m}} for some m.

Let f(x, y) ∈ C[[x, y]] be a formal power series without constant term. We seek series y(x) without constant term such that f(x, y(x)) = 0.

Let us first eliminate a marginal case; if f(0, y) = 0, it means that f(x, y) is divisible by some power of x; let a be the maximum power of x dividing f(x, y), and let us set f(x, y) = x^{a}f′(x, y). Geometrically, the equality f(0, y) = 0 means that the curve f(x, y) = 0 contains the y-axis, and the equality above means that this axis should be counted a times in the curve. This component may be parametrized by x = 0, y = t and we are left with the problem of parametrizing the rest of the curve, which is defined by f′(x, y) = 0. We now have f′(0, y) ≠ 0, and we may thus reduce to the case f(0, y) ≠ 0. From now on we shall assume that f(0, y) ≠ 0. We may then write, since f(0, y) is a formal power series in y, f(0, y) = y^{n}g(y), with g(0) ≠ 0.

The proof of the existence of parametrizations proceeds by induction on the integer n. If n = 1, we have \frac{∂f}{∂y} (0, 0) ≠ 0, and by the implicit function

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theorem there exists a unique formal power series y(x) ∈ C[[x]] such that y(0) = 0 and f(x, y(x)) = 0. We now assume that n > 1. Considering series f(x, y) of the form y^{n} − x^{q} with n, q > 1 and (n, q) = 1 shows that one cannot hope to find series in powers of x. Newton’s idea is to seek solutions which are fractional power series in x, that is, he seeks series in x^{\frac{1}{m}} for some integer m, say ϕ(x^{\frac{1}{m}}) ∈ C[[x^{\frac{1}{m}}]] such that f(x, ϕ(x^{\frac{1}{m}})) = 0. More precisely he seeks solutions of the form:

y = x^{ν}(c_{0} + ϕ_{0}(x^{\frac{1}{m}}))

with c_{0} ≠ 0, ν ∈ Q_{+}, ϕ_{0} without constant term. If we write

f(x, y) = ∑_{i,j∈N} a_{i,j}x^{i}y^{j} with a_{0,0} = 0

and substitute, we get

∑_{i,j} a_{i,j}x^{i+νj}(c_{0} + ϕ_{0}(x^{\frac{1}{m}}))^{j}

and we seek ν, c_{0} ≠ 0 and a series ϕ_{0}(x^{\frac{1}{m}}) such that this series is zero. In particular, its lowest order terms in x must be zero. Since ϕ_{0} has no constant term, if we denote by μ the minimum value of i + νj for (i, j) such that a_{i,j} ≠ 0, we have

∑_{i,j} a_{i,j}x^{i+νj}(c_{0} + ϕ_{0}(x^{\frac{1}{m}}))^{j} = x^{μ} ∑_{i+νj=μ} a_{i,j}c_{0}^{j} + x^{μ}h(x^{\frac{1}{m}})

where h has no constant term. So c_{0} must satisfy

∑_{i+νj=μ} a_{i,j}c_{0}^{j} = 0

For this equation to have a non-zero root in C, it is necessary and sufficient that the sum has more than one term.

Let us consider in the (i, j)-plane the set of points (i, j) such that a_{i,j} ≠ 0. It is a subset N (f) of the first quadrant

R_{0}^{2} = {(i, j) /i ≥ 0, j ≥ 0},

called the Newton cloud of the series f(x, y). Any two subset A and B of R^{d} can be added coordinate-wise, to give the Minkowski sum A+B = {a+b, a ∈ A, b ∈ B} of A and B. Let us consider the subset N_{+}(f) = N (f) + R_{0}^{2} of R_{0}^{2}; its boundary is a sort of staircase with possibly infinite vertical or horizontal parts. The Newton polygon P(f) of f(x, y) is defined as the boundary of the convex hull of N_{+}(f). It is a broken line with infinite horizontal and vertical sides, possibly different from the coordinate axis.

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Just

above is a picture of a Newton polygon in the case where the infinite sides do coincide with the coordinate axis, or equivalently where the area bounded by the polygon is finite.

Recall that the convex hull of a subset of R^{d} can be defined as the intersection of the half-spaces which contain it. A half-space is the set of points situated on one side of an affine hyperplane. Thus, the number

μ = min_{a_{i,j}≠0}{i + νj}

is the minimal abscissa of the intersection points with the i-axis of the lines with slope \frac{−1}{ν} meeting N_{+}(f). Let us denote by L_{ν} the line which gives this minimum; an example in drawn on the picture. So the polynomial

∑_{i+νj=μ} a_{i,j}c_{0}^{j}

corresponds to the sum of the terms a_{i,j}x^{i}y^{j} such that (i, j) lies on the intersection of the line L_{ν} with the Newton polygon. A necessary and sufficient condition for this polynomial to have more than one term is that \frac{−1}{ν} is the slope of one of the sides of the Newton polygon. For simplicity of notation, let us call ν the inclination of the line of slope \frac{−1}{ν}. Let us denote by \frac{l}{h} the inclination of the ”first side” of the Newton polygon of f, that is, the side with the smallest inclination. Let c_{0} be a non zero root of the corresponding equation, and let us make the change of variables

\begin{matrix}x = & x_{1}^{l} \\ y = & x_{1}^{h}(c_{0} + y_{1}) \end{matrix}

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The substitution in f(x, y) gives

f(x_{1}^{h}, x_{1}^{ℓ}(c_{0} + y_{1})) = ∑ a_{i,j}x_{1}^{hi+ℓj}(c_{0} + y_{1})^{j} .

By definition of μ, for each a_{i,j} ≠ 0, we have hi+ℓj ≥ μh, so we may factor the series above as

x_{1}^{μh}f_{1}(x_{1}, y_{1}) , where f_{1}(x_{1}, y_{1}) = ∑ a_{i,j}x_{1}^{hi+ℓj−μh}(c_{0} + y_{1})^{j} .

We remark that

f_{1}(0, y_{1}) = ∑_{i+νj=μ} a_{i,j}(c_{0} + y_{1})^{j} ,

and since a_{0,k} ≠ 0 by definition of n, the order in y_{1} of f_{1}(0, y_{1}) is ≤ n.

Since c_{0} has been chosen as a root of the polynomial ∑_{i+νj=μ} a_{i,j}c_{0}^{j}, this order is ≥ 1. We remark that The order in y_{1} of f_{1}(0, y_{1}) is equal to n if and only if c_{0} is a root of multiplicity n of the polynomial ∑_{i+νj=μ} a_{i,j}T^{j} = 0 But then we must have an equality

∑_{i+νj=μ} a_{i,j}T^{j} = a_{0,n}(T − c_{0})^{n}

which implies by the binomial formula and since C is a field of characteristic zero, that the term in T^{n−1} is not zero; this is possible only if ν is an integer and then the equality above shows that the ”first side of the Newton polygon” meets the horizontal axis at the point (νn, 0), which corresponds to the monomial x^{νn}, which has the non zero coefficient (−1)^{n}a_{0,n}c_{0}^{n}, so it is actually the only finite side of the Newton polygon of f(x, y), which means that we may write in this case

f(x, y) = a_{0,n}(y − c_{0}x^{ν})^{n} + ∑_{i+νj>μ} a_{i,j}x^{i}y^{j} with ν ∈ N, μ = νn.

Making the change of variables

\begin{matrix}x = & x_{1} \\ y = & y_{1} + c_{0}x_{1}^{ν} \end{matrix}

the series f(x, y) becomes

f′(x, y) = a_{0,k}y_{1}^{n} + ∑_{i+νj>μ} a_{i,j}x_{1}^{i}(y_{1} + c_{0}x_{1}^{ν})^{j}

The monomials which appear are of the form x_{1}^{i+νl}y_{1}^{j−l}, so that they all satisfy i + νl + ν(j − l) = i + νj > νn. This means that if the order

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of f_{1}(0, y_{1}) is n, the Newton polygon of f_{1}(x_{1}, y_{1}) still contains the point (0, n) and the inclination ν_{1} of its first side is strictly greater than ν.

The proof now proceeds as follows, :

a) If the order in y_{1} of f_{1}(0, y_{1}) is less than n, by the induction hypothesis, there exist an integer m_{1} and a series ϕ_{1}(x1^{\frac{1}{m_{1}}} ) ∈ C[[x1^{\frac{1}{m_{1}}} ]] such that

f_{1}(x_{1}, ϕ_{1}(x1^{\frac{1}{m_{1}}} )) = 0

By the definition of f_{1}, this implies that

f(x_{1}^{h}, x_{1}^{ℓ}(c_{0} + ϕ_{1}(x1^{\frac{1}{m_{1}}} )) = 0

If we set m = m_{1}h and ϕ(x^{\frac{1}{m}}) = x^{\frac{l}{h}} (c_{0} + ϕ_{1}(x^{\frac{1}{m}})) ∈ C[[x^{\frac{1}{m}}]], we have f(x, ϕ(x^{\frac{1}{m}})) = 0 and the result in this case.

b) If the order in y_{1} of f(0, y_{1}) is still equal to n, we saw that ν is an integer and the inclination of the first side of the Newton polygon of the function f_{1}(x_{1}, y_{1}) obtained from f(x, y) as above is strictly greater than ν.

We now set ν_{0} = ν ∈ N and repeat the same analysis for f_{1}, defining a function f_{2}(x_{2}, y_{2}). If again the order of f_{2}(0, y_{2}) is n, the slope of the first side of the Newton polygon of f_{1}(x_{1}, y_{1}) is an integer ν_{1} > ν_{0} and after the change of variables x = x_{2}, y = y_{2} + c_{0}x_{2}^{ν_{0}} + c_{1}x^{ν_{1}} the slope of the Newton polygon has become greater than ν_{1}.

There are two possibilities;

– either after a finite number of such steps we get a function f_{p}(x_{p}, y_{p}) such that f(0, y_{p}) is of order < n, and by the induction hypothesis we have a series ϕ_{p}(x^{\frac{1}{mp}}) ∈ C[[x^{\frac{1}{mp}}]] such that f_{p}(x, ϕ_{p}(x^{\frac{1}{mp}})) = 0, and so a series

y = c_{0}x^{ν_{0}} + c_{1}x^{ν_{1}} + ⋯ + c_{p−1}x^{ν_{p−1}} + ϕ_{p}(x^{\frac{1}{mp}})

such that f(x, y(x)) = 0;

Or the order remains indefinitely equal to n and we have an infinite increasing sequence of integers

ν_{0} < ν_{1} < … < ν_{p} < …

and a formal power series

ϕ_{∞}(x) = c_{0}x^{ν_{0}} + c_{1}x^{ν_{1}} + ⋯ + c_{p}x^{ν_{p}} + ⋯ ∈ C[[x]]

such that the Newton polygon of the function f_{∞}(x_{∞}, y_{∞}) obtained from f(x, y) by the change of variables x = x_{∞} , y = y_{∞} + ϕ_{∞}(x) has a Newton polygon containing the point (0, n) and with inclination 0. This means that f_{∞}(x_{∞}, y_{∞}) is divisible by y_{∞}^{n}, so we may write

f_{∞}(x_{∞}, y_{∞}) = y_{∞}^{n}g(x_{∞}, y_{∞})

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This implies that the order of g(0, y_{∞}) is zero, so g(0, 0) ≠ 0. Geometrically, our curve is the non singular curve y = ϕ_{∞}(x) counted n times. Indeed, for each integer p, we have

f(x, c_{0}x^{ν_{0}} + c_{1}x^{ν_{1}} + ⋯ + c_{p}x^{ν_{p}}) = x^{ν_{0}+ν_{1}+···+ν_{p}}f_{p}(x, 0),

so that by Taylor’s expansion theorem, f(x, ϕ(x)) = 0. This completes in the formal case the proof of the existence of a fractional power series such that f(x, y(x)) = 0.

In order to describe all the solutions of the equation f(x, y) = 0, it is convenient to develop a little more the formalism of the Newton polygon. Let P and P′ be two Newton polygons; we can define their sum P + P′ as the boundary of the convex hull of the Minkowski sum of the convex domains in R_{+}^{2} bounded by P and P′ respectively. It is easy to verify that the following equality holds for f, f′ ∈ C[[x, y]]

P(ff′) = P(f) + P(f′).

Any Newton polygon has a length and an height which are the length of the horizontal and vertical projections of its finite part, respectively.

We say that a Newton polygon is elementary if it has only one finite side. If it bounds a finite area, it is then uniquely determined by its length and height. We use the following notation for such an elementary Newton polygon.

We also need a little more algebra, beginning with the following fundamental theorem: One says that a holomorphic function f(x_{1}, …, x_{d}, y) defined on a neighborhood of 0 in C^{d} ×C is y-regular (of order n) if f(0, y) has a zero of finite order n at 0 ∈ {0} × C. Geometrically this means that if we consider the germ of hypersurface (W, 0) ⊂ C^{d} × C defined by f(x_{1}, …, x_{d}, y) = 0 and the first projection p: W → C^{d}, then for a small enough representative, if W is not empty (i.e., n ≥ 1), the fiber p^{−1}(0) is the single point 0. In other words, the fiber is a finite subset of {0}×C. The general idea of the avatars

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of the Weierstrass preparation theorem is that finiteness of the fiber over one point x in an analytic map implies finiteness of the fibers above points sufficiently close to x.

Theorem 3.1. (Weierstrass preparation Theorem) If f(x_{1}, …, x_{d}, y) is regular of order n in y, there exist a unique polynomial of the form

P (x_{1}, …, x_{d}, y) = y^{n} + a_{1}(x_{1}, …, x_{d})y^{n−1} + ⋯ + a_{n}(x_{1}, …, x_{d})

with a_{i} ∈ C{x_{1}, …, x_{d}} and a convergent series u(x_{1}, …, x_{d}) with u(0) ≠ 0, i.e., invertible in C{x_{1}, …, x_{d}} such that we have the equality of convergent series

f(x_{1}, …, x_{d}, y) = u(x_{1}, …, x_{d}, y)P (x_{1}, …, x_{d}, y).

The polynomial P is said to be distinguished in y, or to be a Weierstrass polynomial.

If we start with any power series f, we have the same result but in the ring of formal power series. It can be shown that, given a function f, for almost every choice of coordinates in C^{n} ×C, the function f is distinguished with respect to the last coordinate.

It follows from the Weierstrass preparation theorem that provided we have chosen coordinates such that f(0, y) ≠ 0, say f(0, y) = a_{q}y^{q} +⋯ with a_{q} ≠ 0, it is equivalent to seek solutions of f(x, y) = 0 and of P (x, y) = 0, where P (x, y) is the Weierstrass polynomial

u^{−1}(x, y)f(x, y) = y^{q} + a_{1}(x)y^{q−1} + ⋯ + a_{q}(x) = 0 with a_{i}(x) ∈ C[[x]]

Now from an algebraic point of view, we must consider the field of fractions C((x)) of the integral domain C[[x]]; the irreducible polynomial T^{m} − x ∈ C((x))[T] defines an algebraic extension of degree m of C((x)), denoted by C((x^{\frac{1}{m}})) which is a Galois extension with Galois group equal to the group μ_{m} of m-th roots of unity in C. The action of μ_{m} is exactly the change in determination of x^{\frac{1}{m}}, determined by x^{\frac{1}{m}} ↦ ωx^{\frac{1}{m}} for ω ∈ μ_{m}. A series of the form y = ∑ a_{i}x^{\frac{i}{m}} such that the greatest common divisor of m and all the exponents i which effectively appear is 1 gives m different series as ω runs through μ_{m}.

Suppose now that our function f is an irreducible element of C[[x, y]], and that the order in y of f(0, y) is n < ∞. Then the construction described

14 Bernard Teissier

above provides a series y(x^{\frac{1}{m}}) with m ≤ n such that f(x, y(x^{\frac{1}{m}})) = 0. In fact m = n since f is irreducible.The product

Π_{ω∈μ_{m}}(y − y(ωx^{\frac{1}{m}}))

is a polynomial Q(x, y) ∈ C[[x]][y] which, by the algorithm of division of polynomials in C((x))[y], divides P (x, y); the rest of the division of P by Q is a polynomial of degree < n − 1 with n different roots; it is zero. We have therefore Q(x, y) = P (x, y) and m = n in this case. We remark that the expansions y(ωx^{\frac{1}{n}}) all have the same initial exponent \frac{l}{h}, here \frac{l}{n}, and by the definition of Q(x, y), only monomials x^{i}y^{j} with \frac{i}{ℓ} + \frac{j}{h} ≥ \frac{μ}{ℓ} appear, and the monomial x^{h} actually appears. So we have verified:

Proposition.- The Newton polygon of an irreducible series is elementary, and of the form \{ \frac{p}{\overline{n}} \}, where n is the order of f(0, y).

Now it is known that rings such as k[[x, y]], where k is a field, or C{x, y} are factorial ; each element has a decomposition f = f_{1}^{a_{1}} … f_{r}^{a_{r}} where f_{i} is irreducible, which means that it cannot be factored again as a product f_{i} = gh in a non trivial way, that is, without g or h being an invertible element in k[[x, y]], (= a series with a non zero constant term). My aim now is to prove the following

Theorem 3.2. .- a) Let k be an algebraically closed field of characteristic zero, and let f ∈ k[[x, y]] be a power series without constant term such that f(0, y) ≠ 0. Consider the decomposition f = uf_{1}^{a_{1}} … f_{r}^{a_{r}} of f into irreducible Weierstrass polynomials f_{i}^{a_{i}}, with a factor u which is invertible in k[[x, y]]. For each index i, 1 ≤ i ≤ r, there are power series without constant term x_{i}(t), y_{i}(t) ∈ k[[t]] such that f(x_{i}(t), y_{i}(t)) ≡ 0; we may choose x_{i}(t) = t^{m_{i}} where m_{i} is the degree of the Weierstrass polynomial f_{i}, and y_{i}(t) is then uniquely determined. Moreover if we then write y_{i}(t) = c_{i}t^{l_{i}} + … with c_{i} ∈ k^{∗}, then the Newton polygon of f in the coordinates (x, y) is the sum

N (f) = ∑_{1}^{r}\{ \frac{a_{i}m_{i}}{\overline{a_{i}l_{i}}} \}.

Here we have to allow the case where for some i, y_{i}(t) ≡ 0, that is l_{i} = ∞.

b) If k = C and f ∈ C{x, y} is a convergent power series, the series x_{i}(t) and y_{i}(t) are also convergent.

Complex curve singularities 15

Remark: if we do not assume f(0, y) ≠ 0, a similar result holds, but we may no longer apply Weiertrass’ theorem and we have to allow expansions of the form x = 0, y = t and the corresponding Newton polygons appears as summands in N (f).

The geometric interpretation of this result is that if we take any reduced analytic plane curve f = uf_{1} … f_{r} with f_{i} irreducible, i.e., all a_{i} = 1, the curve defined by f = 0 is a sufficiently small neighborhood of the origin is the analytic image of a representative of a complex-analytic map-germ

⨆_{i=1}^{r}(C, 0)_{i} ⟶ (C^{2}, 0)

which we can explicitely build by using Newton’s method. Remark The Newton polygon depends upon the coordinates. One usually chooses the coordinates (x, y) in such a way that the degree of the Weierstrass polynomial is equal to the order of the equation f(x, y). I leave it as an exercize to show that if one writes the series f as a sum of homogeneous polynomials

f(x, y) = f_{n}(x, y) + f_{n+1}(x, y) + ⋯,

where f_{i} is homogeneous of degree i, this condition is equivalent to: f_{n}(0, y) ≠ 0.

Conversely, given two power series x(t), y(t) ∈ k[[t]] without constant term, one may eliminate t between them to produce an equation f(x, y) = 0 with the property that f(x(t), y(t) = 0. Indeed, by using the ”natural” elimination process (see[T1]) we may do this in such a way that eliminating t beween x(t^{q}), y(t^{q}) produces the equation f^{q}(x, y), so that we may even represent parametrically a non-reduced equation.

There are several ways to prove this theorem; one is to prove the convergence first, either directly by providing bounds for the coefficient of the series produced by Newton’s method, which works but is inelegant, or by considering the analytic curve f(t^{m}, y) = 0, and proving that it is a ramified analytic covering of the t-axis; it is also the union of m non singular curves, so each of them is analytic, and this proves the convergence of the series. (see [L], II.6).

These proofs give no basis for generalizations to higher dimension, so I chose to present a geometric method of constructing the analytic map

⨆_{i=1}^{r}(C, 0)_{i} ⟶ (C^{2}, 0).

16 Bernard Teissier

This method was perfected by Hironaka and is the basis for his method of resolution in all dimensions over a field of characteristic zero. Remark In the study of analytic functions of one variable near one of their zeroes, a basic fact is that given two monomials x^{a}, x^{b}, one must divide the other in C{x}, C[[x]], or even C[x]. This allows us to write any series f(x) = x^{a}u(x) with u(0) ≠ 0, in C{x}, and the local behavior of f is determined by the integer a. It is no longer true that given two monomials in (x, y), one must divide the other; the typical example is the pair of monomials y^{n}, x^{q}. In particular, the ideal of C{x, y} generated by all the monomials appearing in the expansion f(x, y) = ∑ a_{ij}x^{i}y^{j} is no longer principal. However, since C{x, y} is a nœtherian ring, this ideal is finitely generated. If we plot the quadrant R_{ij} = (i, j) + R_{+}^{2} for each monomial x^{i}y^{i} appearing in our series, and observe that the integral points in this quadrant correspond to the monomials which are multiples of x^{i}y^{j}, we have a graphic way of representing the generators of the ideal generated by all the monomials appearing in the series f: Consider the union of all the R_{ij} for (i, j)/a_{ij} ≠ 0; its boundary is a sort of staircase. Our generators correspond to the insteps of the staircase. The convex hull of the union is the Newton polygon. Note finally that from the viewpoint of considering lines L_{ν} : i + jν = c as above, and where they meet the staircase, it is the convex hull which is relevant.

Remark I The Newton-Puiseux theorem is strictly a characteristic zero fact. It implies in particular that if a fractional power series in x is a solution of an algebraic equation with coefficients in C[x] or C{x} or C[[x]], the denominators of the exponents of x appearing in that power series are bounded. Let k be a field of characteristic p, and consider the series where

Complex curve singularities 17

the exponents have unbounded denominators:

y = ∑_{i=1}^{∞} x^{1− \frac{1}{p^{i}}};

it is a solution of the algebraic equation

y^{p} − x^{p−1}(1 + y),

as one can check directly. It is an Artin-Schreier equation. Exercise Find where the proof of Newton’s theorem uses characteristic zero.

The Newton-Puiseux theorem in the formal case is equivalent to the statement that the algebraic closure of the field of fractions k((x)) of the ring k[[x]] of formal power series in one variable is equal to ⋃_{n=1}^{∞} k((t^{\frac{1}{n}})) when the field k is algebraically closed of characteristic zero. The corresponding result in positive characteristic has been proved recently by Kiran S. Kedlaya (see [Ke]) and is quite a bit more delicate.

4. Puiseux exponents

Let f(x, y) ∈ C{x, y} be such that f(0, y) = y^{n}u(y) with u ∈ C{y}, o(0) ≠ 0. As we have just seen, it is equivalent to find solutions y(x) for f and to find roots of the Weierstass polynomial

P (x, y) = y^{n} + a_{1}(x)y^{n−1} + ⋯ + a_{n}(x)

corresponding to f. If the element f(x, y) ∈ C{x, y} is irreducible, so is the Weierstrass polynomial in C{x}[y]. Newton’s theorem tells us that such an irreducible polynomial has all its roots of the form

y = ∑_{i=1}^{∞} a_{i}ω^{i}x^{\frac{i}{n}}

where ω runs through the n-th roots of unity in C.

This is equivalent to the statement that an analytically irreducible curve as above can be parametrized in the following manner:

\begin{matrix}x = & t^{n} \\ y = & ∑_{i=1}^{∞} a_{i}t^{i} \end{matrix}

In particular, this shows that the polynomial P determines a Galois extension of the field of fractions C{{x}} of C{x}, and of the field of fractions C((x)) of C[[x]], with Galois group μ_{n}.

18 Bernard Teissier

A direct consequence of this is the:

Theorem 4.1. (Newton-Puiseux Theorem). The algebraic closure of the field C\{\{x\}\} (resp. C((x)) ) is the field ⋃_{n≥1} C\{\{x^{\frac{1}{n}} \}\} (resp. ⋃_{n≥1} C((x^{\frac{1}{n}})) )

This result is the algebraic counterpart of the fact that the fundamental group of a punctured disk is Z. A linear projection onto (C, 0) of small representative of a germ of complex analytic curve (X, 0) ⊂ (C^{2}, 0) which is finite (the curve does not contain the kernel of the linear projection as one of its components) can be restricted over a small punctured disk D_{η}^{∗} = D_{η} \ {0} to give a finite covering of D_{η}^{∗}, which is connected for all sufficiently small η if and only if (X, 0) is analytically irreducible at 0. It can be shown that conversely the connected coverings of a punctured disk correspond to irreducible curves as above, and that in this correspondence, the Galois group of the covering, which is of the form Z/kZ since the fundamental group of the disk is Z, corresponds to the Galois group of the extension of the field C{{x}} or of the field C((x)) defined as above by the curve.

Let

f(x, y) = 0 with f(x, y) ∈ C{x, y}

be an equation for a branch (X, 0) ⊂ (C^{2}, 0), which means that the series f is an irreducible element of C{x, y}.

As we saw, we may assume thanks to the Weierstrass preparation theorem that f is of the form

f(x, y) = y^{n} + a_{1}(x)y^{n−1} + ⋯ + a_{n}(x)

where n is the intersection multiplicity at 0 of the branch C with the axis x = 0.

As we saw, after possibly a change of coordinates to achieve that x = 0 is transversal to it at 0,the branch X can be parametrized near 0 as follows

\begin{aligned}x(t) & = t^{n} \\ y(t) & = a_{m}t^{m} + a_{m+1}t^{m+1} + ⋯ + a_{j}t^{j} + ⋯ with m ≥ n \end{aligned}

Let us now consider the following grouping of the terms of the series y(t): set β_{0} = n and let β_{1} be the smallest exponent appearing in y(t) which is not divisible by β_{0}. If no such exponent exists, it means that y is a power series in x, so that our branch is analytically isomorphic to C, hence non

Complex curve singularities 19

singular. Let us suppose that this is not the case, and set e_{1} = (n, β_{1}), the greatest common divisor of these two integers. Now define β_{2} as the smallest exponent appearing in y(t) which is not divisible by e_{1}. Define e_{2} = (e_{1}, β_{2}); we have e_{2} < e_{1}, and we continue in this manner. Having defined e_{i} = (e_{i−1}, β_{i}), we define β_{i+1} as the smallest exponent appearing in y(t) which is not divisible by e_{i}. Since the sequence of integers

n > e_{1} > e_{2} > ⋯ > e_{i} > ⋯

is strictly decreasing, there is an integer g such that e_{g} = 1. At this point, we have structured our parametric representation as follows:

x(t) = t^{n}
y(t) = a_{n}t^{n} + a_{2n}t^{2n} + ⋯ + a_{kn}t^{kn}+
+a_{β_{1}}t^{β_{1}} + a_{β_{1}+e_{1}}t^{β_{1}+e_{1}} + ⋯ + a_{β_{1}+k_{1}e_{1}}t^{β_{1}+k_{1}e_{1}}

+a_{β_{2}}t^{β_{2}} + a_{β_{2}+e_{2}}t^{β_{2}+e_{2}} + ⋯ + a_{β_{q}}t^{β_{q}} + a_{β_{q}+e_{q−1}}t^{β_{q}+e_{q−1}} + ⋯

+a_{β_{g}}t^{β_{g}} + a_{β_{g}+1}t^{β_{g}+1} + ⋯

where by construction the coefficients of the t^{β_{i}}; i ≥ 1 are not zero. Let us define integers n_{i} and m_{i} by the equalities

e_{i−1} = n_{i}e_{i}, β_{i} = m_{i}e_{i} for 1 ≤ i ≤ g

and note that we may rewrite the expansion of y into powers of t as an expansion of y into fractional powers of x as follows:

y = a_{n}x + a_{2n}x^{2} + ⋯ + a_{kn}x^{k}+
+a_{β_{1}}x^{\frac{m_{1}}{n_{1}}} + a_{β_{1}+e_{1}}x^{\frac{m_{1}+1}{n_{1}}} + ⋯ + a_{β_{1}+k_{1}e_{1}}x^{\frac{m_{1}+k_{1}}{n_{1}}}

+a_{β_{2}}x^{\frac{m_{2}}{n_{1}n_{2}}} + a_{β_{2}+e_{2}}x^{\frac{m_{2}+1}{n_{1}n_{2}}} + ⋯ + a_{β_{q}} x^{\frac{mq}{n_{1}n_{2}···nq}} + a_{β_{q}+e_{q−1}}x^{\frac{mq +1}{n_{1}n_{2}···nq}} + ⋯

+a_{β_{g}} x^{\frac{mg}{n_{1}n_{2}···ng}} + a_{β_{g}+1}x^{\frac{mg +1}{n_{1}n_{2}···ng}} + ⋯

The set of pairs of coprime integers (m_{i}, n_{i}) are sometimes also called the Puiseux characteristic pairs. Their datum is obviously equivalent to that of the characteristic exponents β_{i}. The sequence of integers B(X) = (β_{0}, β_{1}, …, β_{g}), where β_{0} = n, may be characterized algebraically as follows: let μ_{n} denote the group of n-th roots of unity. For ω ∈ μ_{n} let us compute the order in t of the series y(t) − y(ωt). If we write ω = e^{\frac{2πik}{n}}, we have

y(ωt) = a_{n}ω^{n}t^{n} + ⋯ + a_{β_{1}}ω^{β_{1}}t^{β_{1}} + ⋯

and we see that multiplying t by ω does not affect the terms in t^{jn}. The term in t^{β_{1}} is unchanged if and only if ω^{β_{1}} = 1, that is \frac{kβ_{1}}{n} is an integer, i.e., kβ_{1} = ln or km_{1} = ln_{1} with the notations introduced above. Since

20 Bernard Teissier

n_{1} and m_{1} are coprime, this means that k is a multiple of n_{1}, which is equivalent to saying that ω belongs to the subgroup μ_{\frac{n}{n_{1}}} of μ_{n} consisting of \frac{n}{n_{1}} = n_{2} ⋯ n_{g}-th roots of unity. If this is the case, then the coefficients of all the terms of the form t^{β_{1}+je_{1}} in the Puiseux expansion are also unchanged when t is multiplied by ω, and the first term which may change is a_{β_{2}}t^{β_{2}}. An argument similar to the previous one shows that if ω ∈ μ_{\frac{n}{n_{1}}} , then ω^{β_{2}} = 1 if and only if ω ∈ μ_{\frac{n}{n_{1}n_{2}}} , and so on.

Finally, if we denote by v the order in t of an element of C{t}, we see that

v(y(t) − y(ωt)) = β_{i} if and only if ω ∈ μ_{\frac{n}{n_{1}···n_{i−1}}} \ μ_{\frac{n}{n_{1}···n_{i}}} for 1 ≤ i ≤ g

This provides an algebraic characterization, and a sequence of cyclic subextensions

C\{x\} ⊂ C\{x^{\frac{1}{n_{1}}} \} ⊂ C\{x^{\frac{1}{n_{1}n_{2}}} \} ⊂ ⋯ ⊂ C\{x^{\frac{1}{n_{1}n_{2}···n_{i}}} \} ⊂ ⋯ ⊂ C\{x^{\frac{1}{n}} \}

corresponding to the nested subgroups μ_{\frac{n}{n_{1}···n_{i}}} of th group μ_{n}.

This shows that the sequence (β_{0}, β_{1}, … β_{g}) depends only upon the ring inclusion C{x} ⊂ O_{X,0}.

We shall see later in a different way that this sequence does not depend upon the choice of coordinates (x, y) in which we write the Puiseux expansion as long as the curve x = 0 is transversal to X. If this is not the case, one still obtains other characteristic exponents, which are related to the transversal ones by the inversion formula which I leave as an exercise (or see [PP] and [GP]). As an example consider the curve with equation y^{3} − x^{2} = 0.

5. From parametrizations to equations

We have just seen an algorithm to produce local parametrizations of the branches of a complex analytic plane curve from its equation. To go in the other direction is to eliminate for each parametrized branch the variable t between the equations x − x(t) = 0, y − y(t) = 0, and then make the product of the equations obtained. Elimination is in general computationally arduous. Is this special case, we have a direct method as follows: write our parametrization in the form x = t^{n}, y = ξ(t) = ∑_{i} a_{i}t^{i}. The product

Π_{ω∈μ_{n}}(y − ξ(ωt))

is invariant under the action of μ_{n} by t ↦ ωt; it is a series f(t^{n}, y)) = f(x, y) which has the property that f(x, y) = 0 is an equation for our curve.

Complex curve singularities 21

However this method does not work for curves in 3-dimensional space, or in positive characteristic. Here is my favourite method (see [T5]) to compute images, explained in this special case.

5.1. Fitting ideals

Let M be a finitely generated module over a commutative nœtherian ring A; then we have a presentation, which is an exact sequence of A-modules

A^{q} → A^{p} → M → 0

The map A-linear map A^{q} → A^{p} is represented, in the canonical basis, by a matrix with entries in A. For each integer j, consider the ideal F_{j}(M) of A generated by the (p − j) × (p − j) minors of that matrix. Note that if j ≥ p, then F_{j}(M) = A (the empty determinant is equal to 1), and if p − j > q, then F_{j}(M) = 0 (the ideal generated by the empty set is (0)). It is not very difficult to check that F_{j}(M) depends only on the A-module M, and not on the choice of presentation. Moreover, if A → B is a morphism of commutative rings, the sequence

B^{q} → B^{p} → M ⊗_{A} B → 0

is a presentation of the B-module B ⊗_{A} M, with the same matrix; therefore F_{j}(M ⊗_{A} B) = F_{j}(M).B. One says that The formation of Fitting ideals commutes with base change.

The most important feature of Fitting ideals, is as follows:

Proposition 5.1. A maximal ideal m of A contains F_{j}(M) if and only if

dim_{A/m}M ⊗_{A} A/m > j.

Proof. Tensoring with A/m the presentation of M gives for each maximal ideal of A an exact sequence of A/m-vector spaces

(A/m)^{q} → (A/m)^{p} → M ⊗_{A} A/m → 0.

the dimension of the cokernel is > j if and only if the rank of the matrix describing the map (A/m)^{q} → (A/m)^{p} is < p − j, which means that all the p − j minors are 0 modulo m, which means that F_{j}(M) ⊂ m.

Let me explain what this has to do with elimination: suppose that we have a branch parametrized by x(t), y(t). This gives a map C{x, y} → C{t}. Observe that this map of C-algebras gives C{t} the structure of a finitely

22 Bernard Teissier

generated C{x, y}-module. Indeed, since x ↦ t^{n} say, it is even a finitely generated C{x}-module, generated by (1, t, …, t^{n−1}). We can therefore write a presentation of C{t} as C{x, y}-module:

C{x, y}^{q} → C{x, y}^{p} → C{t} → 0.

Now it is a theorem of commutative algebra that since C{x, y} is a 2dimensional regular local ring, for every finitely generated C{x, y}-module M, if we begin to write a free resolution by writing M as a quotient of a finitely generated free C{x, y}-module, C{x, y}^{p} → M, then writing the kernel of that map as a quotient of a finitely generated free C{x, y}-module, and so on, this has to stop after 2 steps. This means that the kernel above is already free, so that in fact, in our case, we have an exact sequence

0 → C{x, y}^{q} →^{ϕ} C{x, y}^{p} → C{t} → 0.

This immediately implies that we have q ≤ p. On the other hand, the C{x, y}-module C{t} must be a torsion module, which means that it must be annihilated by some element of C{x, y}; intuitively this means that the image of our parametrization has an equation: an element f ∈ C{x, y} such that fC{t} = f(x(t), y(t)) = 0. If this was not the case, there would be an ideal T ⊂ C{t} consisting of the elements which are annihilated by multiplication by some non zero element of C{x, y}. If we assume that T ≠ C{x, y} and remark that by construction our map of algebras C{x, y} → C{t} induces an injection C{x, y} ⊂ C{t}/T, then either T ≠ 0 and we have an injection of C{x, y} in a finite-dimensional vector space over C, which is absurd, or T = 0 and we have an injection C{x, y} ⊂ C{t}, but since C{t} is a finitely generated C{x, y}-module, the two rings should have the same dimension, by the third axiom of dimension theory (see [Ei], 8.1), which is absurd. So C{t} is a torsion C{x, y}-module, which implies that the map induced by ϕ after tensorization of our exact sequence by the field of fractions C{{x, y}} of C{x, y} is surjective, hence q ≥ p and finally q = p.

Now we know that q must be equal to p, and that we have an exact sequence

0 → C{x, y}^{p} ⟶^{ϕ} C{x, y}^{p} → C{t} → 0.

Proposition 5.2. The 0-th Fitting ideal of the C{x, y}-module C{t} is principal and generated by the determinant of the matrix encoding the homomorphism ϕ in the canonical basis.

Complex curve singularities 23

Example 5.1. Consider the parametrization x = t^{2}, y = t^{3}; it makes C{t} into a C{x, y}-module generated by (e_{0} = 1, e_{1} = t). The relations are −ye_{0} + xe_{1}, x^{2}e_{0} − ye_{1}. In this case, p = 2 and the matrix ϕ has entries

(\begin{matrix}−y & x \\ x^{2} & −y \end{matrix})

Exercise: 1) For any integer k, consider the curve parametrized by x = t^{2k}, y = t^{3k}. Show that the Fitting ideal is generated by (y^{2} − x^{3})^{k}.

2) Consider the curve in C^{3} parametrized by x = t^{3}, y = t^{4}, z = 0. In this case, the C{x, y, z}-module C{t} is generated by 1, t, t^{2}. Of course we can no longer hope to have q = p in its presentation, but one can compute a presentation ( see [T5], 3.5.2)

C{x, y, z}^{6} → C{x, y, z}^{3} → C{t} → 0.

and find that the Fitting ideal

F_{0}(C{t}) = (y^{3} − x^{4}, z^{3}, zx^{2}, zy^{2}, z^{2}y, z^{2}x)C{x, y, z}

It defines the plane curve y^{3} − x^{4} = 0, z = 0, plus a 0-dimensional (embedded) component sticking out of the z = 0 plane.

The appearance of this embedded component corresponds to the fact that we can embed our plane curve singularity in a family of space curve singularities, say x = t^{3}, y = t^{4}, z = vt^{5} for example, where v is a deformation parameter. Now if we compute the Fitting ideal of the image of the map C × C → C^{3} × C given by x = t^{3}, y = t^{4}, z = vt^{5}, v = v, and then set v = 0, we must find the Fitting ideal defining the image of our original map C → C^{3}. This is one of the basic properties of Fitting ideals. On the other hand, by a classical result, the embedding dimension in a family of singularities can only increase by specialization. Since for v ≠ 0 the image computed by the Fitting ideal has embedding dimension 3, it must also be 3 for v = 0, so the Fitting ideal must define something which contains our plane curve but has embedding dimension 3. Our zero-dimensional component increases the embedding dimension from 2 to 3.

Now we must prove that the generator of the 0-th Fitting ideal is an acceptable equation of the image of our parametrization.

Given our map π: C{x, y} → C{t} corresponding to the curve parametrization, we could say that the equation of the parametrized curve is given simply by the kernel K of this map of algebras. We are going to prove that the kernel K and the Fitting ideal have the same radical, and so define the same underlying set, but they are not equal in general, and

24 Bernard Teissier

the formation of the kernel does not commute with base extension while the formation of the Fitting ideal does. First, we must check that, with the notations of the definition of Fitting ideals, we have F_{0}(M).M = 0, which means that the Fitting ideal is contained in the kernel. In our case, where q = p, it follows directly from Cramer’s rule if you interpret the statement as: detϕ.C{x, y}^{p} ∈ Image(ϕ). Note that this is true in the general situation of a finitely generated A-module; the Fitting ideal is contained in the annihilator of M. Secondly we must prove that K is contained in the radical of F_{0}(M). Take a non zero element h ∈ K; we have hM = 0, so that applying the rule of base extension to the map A → A[h^{−1}], with A = C{x, y} in this case, we get F_{0}(M)A[h^{−1}] = A[h^{−1}], and since F_{0}(M) is finitely generated, this implies that there exists an integer s such that h^{s} ∈ F_{0}(M), and the result. So we have proved the inclusions

F_{0}(M) ⊆ K ⊆ \sqrt{F_{0}(M)}.

5.2. A proof of Bézout’s theorem (after [T5], §1)

We begin with a Fitting definition of the resultant of two polynomials in one variable. Let A be a commutative ring and

\begin{matrix}P = & p_{0} + p_{1}X + ⋯ + p_{n}X^{n} \\ Q = & q_{0} + q_{1}X + ⋯ + q_{m}X^{m} \end{matrix}

be two polynomials in A[X] Let us assume that p_{n} and q_{m} are invertible in A. The natural ring in which to treat the resultant is

A = Z[p_{0}, …, p_{n}, q_{0}, …, q_{m}, p_{n}^{−1}, q_{m}^{−1}],

considering the two polynomials

\begin{matrix}P = & p_{0} + p_{1}X + ⋯ + p_{n}X^{n} \\ Q = & q_{0} + q_{1}X + ⋯ + q_{m}X^{m} \end{matrix}

now with coefficients in A. The difference is that now the p_{i}, q_{j} have become indeterminates.

Given any ring A and two polynomials P, Q as above with coefficients in A and highest coefficients invertible in A, there is a unique homomorphism ev: A → A such that ev(P) = P, ev(Q) = Q; it sends the indeterminate p_{i} (resp. q_{j}) to the coefficient of X^{i} in P (resp. X^{j} in Q). The A-module A[X]/P is a free A-module of rank n, and multiplication by Q (which is injective since the p_{i}, q_{j} are indeterminates, gives us an exact

Complex curve singularities 25

sequence

0 → A[X]/P →^{ϕ} A[X]/P → A[X]/(P, Q) → 0.

This allows us to compute the 0-th Fitting ideal of the A-module A[X]/(P, Q) as the determinant of the matrix of ϕ.

Definition 5.1. A universal resultant R(P, Q) of the universal polynomials P and Q is a generator with coprime integer coefficients of the 0th Fitting ideal of the A-module A[X]/(P, Q).

Given a ring A and two polynomials as above, the resultant of P and Q is the image ev(R(P, Q) ∈ A. It may be the zero element.

Note that the ring A has a grading given by degp_{i} = n − i, degq_{j} = m − j. If we give X the degree 1, the polynomials P and Q are homogeneous of degree n and m respectively for the corresponding grading of A[X].

In order to deal with graded free modules, it is convenient to introduce the following notation: If A is a graded ring, for any integer e, denote by A(e) the free graded A-module of rank one consisting of the ring A where the degree of an element of degree i in A is of degree i + e in A(e). Any free graded A-module is a sum of A(e_{i}). The proof of Bézout’s theorem relies on the Fitting definition of the resultant and the following two lemmas:

Lemma 5.1. Let A be a graded ring; for any homogeneous homomorphism of degree zero between free graded A-modules

Ψ: ⨁_{i=1}^{p} A(e_{i}) → ⨁_{j=1}^{p} A(f_{j}),

setting M = cokerΨ, the Fitting ideals F_{k}(M) are homogeneous and moreover

degF_{0}(M) = deg(detΨ) = ∑_{i=1}^{p} e_{i} − ∑_{j=1}^{p} f_{j}.

Proof. To say that the morphism is of degree zero means that it sends an homogeneous element to an homogeneous element of the same degree. This implies that the entries of the matrix of Ψ satisfy

degΨ_{ij} = e_{i} − f_{j},

and this suffices to make the minors homogeneous; let us check it for the determinant.

26 Bernard Teissier

Each term in its expansion is a product ψ_{i_{1}j_{1}} … ψ_{i_{p}j_{p}} where each i and j appear exactly once. It is homogeneous of degree

∑ e_{i_{k}} − ∑ f_{j_{k}} = ∑_{i=1}^{p} e_{i} − ∑_{j=1}^{q} f_{j}.

We can now compute the degree of R(P, Q) ∈ A. If we use the presentation given above, we find that the homomorphism ϕ is of degree zero if we give each X^{i} in the first copy of A[X]/P the degree i+m and keep X^{j} of degree j in the second.

Thus we find

Lemma 5.2. We have the equality

degR(P, Q) = ∑_{i=1}^{n}(m + i) − ∑_{j=1}^{n} j = mn.

Remark 5.1. 1) There are other presentations for the A-module A[X]/(P, Q). For example

\begin{aligned}0 → & A[X]/(X^{n}) ⊕ A[X]/(X^{m}) & → A[X]/(P.Q) → A[X]/(P, Q) → 0 \\ & (\overline{a}, \overline{b}) ↦ \overline{aQ + bP} \end{aligned}

or

\begin{matrix}0 → A[X]/(P.Q) → & A[X]/(X^{n}) ⊕ A[X]/(X^{m}) → & A[X]/(P, Q) → 0 \\ \overline{a} & ↦ (\overline{a}, \overline{a}) (\overline{a}, \overline{b}) ↦ & (\overline{a} − \overline{b}) \end{matrix}

The first of these two gives the usual Sylvester determinant, of size m + n. The second follows from from the chinese remainder theorem.

2) The total degree of a polynomial defining an affine plane curve is equal to the degree of the homogeneous polynomial in three variables defining the projective plane curve defining the closure of the affine curve in projective space; it is the degree of the curve.

The other lemma is of the same nature and shows that the Fitting ideal locally computes the image of an intersection of two curve with a multiplicity equal to the intersection multiplicity of the two curves.

Lemma 5.3. Let R be a discrete valuation ring containing a representative of its residue field k, and let v be its valuation. Let

Ψ: R^{p} → R^{p}

be an homomorphism of free R-modules whose cokernel M is of finite length, i.e., a finite-dimensional vector space over k. Then we have the equality

v(detΨ) = dim_{k}M.

Complex curve singularities 27

Proof. A discrete valuation ring is a principal ideal domain. By the main theorem on principal ideal domains we can find bases for both R^{p} such that the matrix representing Ψ is diagonal, with entries a_{1}, …, a_{p} on the diagonal, say. Then clearly v(detΨ) = ∑_{i=1}^{p} v(a_{i}) and dim_{k}M = ∑_{i=1}^{p} dim_{k}R/a_{i}R. Thus it suffices to consider the case where p = 1. Then we have a = uπ^{s} where π is a generator of the maximal ideal of R, and u is invertible in R. Then v(a) = s while R/aR is the k-vector space freely generated by the images of 1, π …, π^{s−1}.

This applies to R = C{t} or R = C[t]_{(t)}, the valuation being the t-adic order.

Now let us begin the proof of Bézout’s theorem. Let A be the graded ring C[x_{1}, x_{2}] and let P, Q ∈ C[x_{0}, x_{1}, x_{2}] be two homogeneous polynomials defining the curves C and D in the complex projective plane, of respective degrees m and n. We can write

\begin{matrix}P = ∑_{i=1}^{n} p_{i}(x_{1}, x_{2})x_{0}^{i} & degp_{i} = n − i \\ Q = ∑_{j=1}^{m} q_{j}(x_{1}, x_{2})x_{0}^{j} & degq_{j} = m − j. \end{matrix}

After a change of coordinates, we may assume that the constants p_{n} and q_{m} are non zero, hence invertible in A. Geometrically this means that the point with homogeneous coordinates (1, 0, 0) does not lie on either of the curves C and D.

As we saw above, there exists a homogeneous morphism of degree zero ev: A → A such that ev(P) = P, ev(Q) = Q, and if the resultant R(P, Q) is not zero, it is of degree mn by Lemma ??. I leave it as an exercise to check, using the factoriality of polynomial rings over C, that R(P, Q) = 0 if and only if C and D have a common component.

Let us now consider the projection π: P^{2}(C) \ (1, 0, 0) → P^{1}(C) given by (x_{0}, x_{1}, x_{2}) ↦ (x_{1}, x_{2}). It induces a well defined projection on C and on D since meither of them contains (1, 0, 0). For each point x ∈ P^{1}(C) there are finitely many points y ∈ C ∩ D such that π(y) = x. By the definition of the resultant and the fact that the formation of the Fitting ideal commutes in particular with localization, we have the following equality:

R(P, Q)O_{P^{1},x} = F_{0}( ⨁_{π(y)=x} O_{P^{2},y}/(P, Q)O_{P^{2},y}).

v_{x}(R(P, Q)) = ∑_{π(y)=x} dim_{C}O_{P^{2},y}/(P, Q)O_{P^{2},y}.

28 Bernard Teissier

Since, if we assume that C and D have no common component, the resultant R(P, Q) is a homogeneous polynomial of degree mn in (x_{1}, x_{2}), it follows from the fundamental theorem of algebra applied to the homogeneous polynomial R that

mn = ∑_{x∈P^{1}} v_{x}(R(P, Q)) = ∑_{y∈C∩D} dim_{C}O_{P^{2},y}/(P, Q)O_{P^{2},y}.

This is Bézout’s theorem if we agree that the intersection multiplicity of C and D at y is equal to

(C, D)_{y} = dim_{C}O_{P^{2},y}/(P, Q)O_{P^{2},y}.

We shall see later that there are many reasons to do that. Finally we get

Theorem 5.1. (Bézout) For closed algebraic curves in P^{2}(C) without common component, we have

degC.degD = ∑_{y∈C∩D}(C, D)_{y}.

6. Resolution of plane curves

Let us consider the projective space P^{n}(C) as the space of lines through the origin in C^{n+1}. If we choose coordinates x_{0}, …, x_{n} on C^{n+1} the projective space is covered by affine charts U_{i}, the points of which correspond to the lines contained in the open set x_{i} ≠ 0. It is customary to take homogeneous coordinates (u_{0} : ⋯: u_{n}) on the projective space, corresponding to the lines given parametrically by x_{i} = u_{i}t, or by the equations x_{i}u_{j} − x_{j}u_{i} = 0, where it is enough to take the n equations for which j = i + 1 and i < n. The term ”homogeneous coordinates ” means that for any λ ∈ k∗ the coordinates (u_{0} : ⋯: u_{n}) and (λu_{0} : ⋯: λu_{n}) define the same point.

Now consider the subvariety Z of the product space C^{n+1} × P^{n} defined by these n equations. It is a nonsingular algebraic variety of dimension n+1 and the first projection induces an algebraic morphism B_{0} : Z → C^{n+1}.

The fiber B_{0}^{−1}(0) is the entire projective space P^{n}(k) since when all x_{i} are zero, all the equations between the u_{j} vanish,while the fiber B_{0}^{−1}(x) for a point x ≠ 0 consists of a unique point because then the coordinates x_{i} determine uniquely the ratios of the u_{j} which means a point of P^{n}(k). Blowing up a point ”replaces the observer at the point by what he sees”, because the observer essentially sees a projective space (in fact a sphere, if we think of a real observer, but this is just a metaphor).

Complex curve singularities 29

A basic properties of blowing up is that it separates lines: in fact consider the algebraic map δ: C^{n+1} \ {0} → P^{n} which to a point outside the origin associates the line joining the origin to this point. Of course we cannot extend the definition of this map through the origin; The graph of δ however, is an algebraic subvariety of (C^{n+1} \ {0}) × P^{n}, and we may take the closure (for the strong topology if k = C, or for the Zariski topology) of this graph. It is a good exercise to check that this closure coincides with Z as defined above. A point of B_{0}^{−1}(0) is precisely a direction of line, so the map δ ∘ B_{0} can be defined there as the map which to this point associates the direction: in Z we have separated all the lines meeting at the origin.

Let us consider in more detail the case n = 1. Then Z is a surface covered by two affine charts corresponding to the charts of the projective space: for convenience of notation set u_{0} = u, u_{1} = v, x_{0} = x, x_{1} = y so that Z is defined by vx − uy = 0. On the open set U of Z where u ≠ 0 we may taxe as coordinates x_{1} = x, y_{1} = \frac{v}{u} and then the map induced by B_{0} on U is described in these coordinates by

\begin{matrix}x ∘ B_{0} = & x_{1} \\ y ∘ B_{0} = & x_{1}y_{1} \end{matrix}

and similarly on the open set V defined by v ≠ 0, we take as coordinates x_{1} = \frac{u}{v}, y_{1} = y and the map B_{0} is described by

\begin{matrix}x ∘ B_{0} = & x_{1}y_{1} \\ y ∘ B_{0} = & y_{1} \end{matrix}

Remark that in the first chart the projective space B_{0}^{−1}(0) is defined by x_{1} = 0 and in the second by y_{1} = 0 (remember that they are coordinates on two distinct charts, and on the intersection of the two charts they define the same subspace). It is a crucial property of blowing up that it transforms the blown-up subspace (here the origin) into a subspace defined locally by one equation (called a divisor); it is a good exercise to check that this is the case in any dimension. The space B_{0}^{−1}(0) is called the exceptionnal divisor. We are now able to study the effect on a function f(x, y) (formal or convergent) of its composition with the map B_{0}. Consider the expansion of f as a sum of homogeneous polynomials

f(x, y) = f_{m}(x, y) + f_{m+1}(x, y) + ⋯ + f_{m+k}(x, y) + ⋯,

where f_{j} is homogeneous of degree j. In the chart U, we may write

\begin{matrix}f ∘ B_{0} = f(x_{1}, x_{1}y_{1}) = \\ x_{1}^{m}(f_{m}(1, y_{1}) + x_{1}f_{m+1}(1, y_{1}) + ⋯ + x_{1}^{k}f_{m+k}(1, y_{1}) + ⋯) \end{matrix}

30 Bernard Teissier

and there is a similar expansion in the other chart. Now if we look at the zero set of f ∘ B_{0} we see that in each chart it contains the exceptionnal divisor counted m times. If we remove this exceptionnal divisor as many times as possible, i.e., divide f ∘ B_{0} by x_{1}^{m} in the first chart and by y_{1}^{m} in the second, we obtain the equation of a curve on the surface Z, either formal or defined near B_{0}^{−1}(0), which no longer contains the exceptionnal divisor. This curve is called the strict transform of the original curve. We also say that the equation obtained in this way is the strict transform of f). In the first chart it is x_{1}^{−m}f(x_{1}, x_{1}y_{1}), and in the second y_{1}^{−m}f(x_{1}y_{1}, y_{1}).

By construction, the strict transform meets the exceptionnal divisor only in finitely many points; let us determine them: in the first chart they are given by f_{m}(1, y_{1}) = 0 and in the second, by f_{m}(1, y_{1}) = 0. By construction of the projective space the points we seek are therefore the points in the projective line defined by the homogeneous equation f_{m}(u, v) = 0. The homogeneous polynomial f_{m} of lowest degree appearing in f(x, y) is called the initial form and f_{m}(x, y) = 0 is a union of m lines (counted with multiplicity) called the tangent cone of f at the point 0. So we see that the strict transform of f meets the exceptionnal divisor at the points in this projective space corresponding to the lines which are in the tangent cone at 0 of our curve.

In particular, if our curve has two components with tangent cones meeting only at the origin, their strict transforms are disjoint. Consider for example f(x, y) = (y^{2} − x^{3})(y^{3} − x^{2}).

In order to analyze in more detail what goes on, we have to assume that k is algebraically closed, which we will do from now on, and introduce the concept of intersection number of two curves at a point. The simplest definition (but not the most useful for computations) is the following: Let f, h ∈ k[[x, y]] be series without constant term and without common irreducible factor. Let (f, h) denote the ideal generated by f and h in k[[x, y]]. Then the dimension

dimk[[x, y]]/(f, h)

is finite and is by definition the intersection number of the two curves at 0. If k = C and f, h are in C{x, y}, then the dimension above is also

dimC{x, y}/(f, h)

where now (f, h) is the ideal generated in C{x, y}.

To prove the finiteness we first remark that it is sufficient to prove it after replacing k by its algebraic closure and then we may use the Hilbert nullstellensatz which tells us that since f = 0, h = 0 meet only at the

Complex curve singularities 31

origin, the ideal (f, h) contains a power of the maximal ideal m = (x, y) say m^{N}. This implies the finiteness since k[[x, y]]/(f, h) is then a quotient vector space of k[[x, y]]/m^{N} and also shows that we may without changing the ideal assume that f, h are polynomials of degree < N, so that for example if f, h are convergent power series the vector spaces C{x, y}/(f, h) and C[[x, y]]/(f, h) are equal.

The definition of intersection multiplicity at the point 0, of the two curves f = 0, h = 0, say in the analytic case is then

(f, h)_{0} = dim_{C}C{x, y}/(f, h).

Note that we use large parentheses for the intersection number, small ones for the ideal generated by f, g.

In any case this definition of the intersection multiplicity has the advantage to suggest the following intuitive interpretation : Consider a 1-parameter deformation of one of the two functions, say f + ϵ; it is possible to show that if f, h converge in a nice neighborhood U of 0, for small enough ϵ, then the two curves h = 0, f +ϵ = 0 meet in U transversally at points which are non-singular on each. Moreover, these points tend to 0 as ϵ tends to 0, and the number of these points is dimC{x, y}/(f, h). So this number may be thought of as the number of ordinary intersections (i.e., transverse intersection of non-singular curves) which are concentrated at 0.

There is another way to present this intersection number, which is very useful for computations: Suppose that h(x, y) = uh_{1}^{e_{1}} … h_{r}^{e_{r}} with u(0) ≠ 0. For each i, 1 ≤ i ≤ r, let us parametrize the curve h_{i}(x, y) = 0 by x(t_{i}), y(t_{i}). Now substitute these power series in f(x, y); we get a series in t_{i}, the order of which we denote by I_{i}. Then we have

I_{i} = dimC{x, y}/(f, h_{i}) ,

and

(f, h)_{0} = ∑_{1}^{r} e_{i}I_{i}.

Remark: Given a germ of curve f = 0, where f = f_{m} + f_{m+2} + ⋯, its multiplicity at the origin may be defined as the smallest degree m of a monomial appearing in the series f. A better definition is to say that the multiplicity is the intersection number (f, ℓ)_{0} for a sufficiently general

32 Bernard Teissier

linear form ℓ. In fact, we have

m ≤ (f, ℓ)_{0}

with equality if and only if the line ℓ(x, y) = 0 is not in the tangent cone defined by f_{m}(x, y) = 0.

Indeed, we may parametrize the line ℓ = 0 by x = αt, y = βt; then we substitute in f:

f(αt, βt) = f_{m}(α, β)t^{m} + f_{m+1}(α, β)t^{m+1} + ⋯

is of order ≥ m, and of order m exactly if and only if f_{m}(α, β) ≠ 0.

It is convenient, given a curve f(x, y) and a point z in the plane, to define the multiplicity of f at z as follows: take coordinates (x′, y,) centered at z, which means that they vanish at z; if z = (a, b) we may take x′ = x−a, y′ = y − b. Then expand f in those coordinates (of course we assume that z is in the domain of convergence of f).

We get f′(x′, y′) = f(a + x′, b + y′). Then we compute the lowest degree terms appearing in the expansion of f′ and denote this by m_{z}(f) or, if X is the curve f(x, y) = 0, by m_{z}(X). We see that m_{z}(f) = 0 unless f(z) = 0, and that if ℓ is a line through z, we have m_{z}(X) ≤ (X, ℓ)_{z} with equality except if ℓ is in the tangent cone of X at z.

Let us apply this, in our blowing up as described above, to the line x_{1} = 0 (the exceptionnal divisor) and the strict transform f_{1}(x_{1}, y_{1}) = 0, at a point x′ with coordinates x_{1} = 0, y_{1} = t_{1}) where f_{m}(1, t_{1}) = 0 i.e., a point of intersection of the strict transform with the exceptionnal divisor. We have

f_{1} = f_{m}(1, y_{1}) + x_{1}f_{m+1}(1, y_{1}) + ⋯

and if we denote by e_{x′} the multiplicity of t_{1} as a root of the polynomial f_{m}(1, Y), it follows from what we saw above that we have

e_{x′} ≥ m_{x′}(f_{1})

with equality unless the curve f_{1}(x_{1}, y_{1}) = 0 is tangent to the exceptionnal divisor at the point x′, in the sense that the tangent at x′ to the exceptionnal divisor is in the tangent cone of f_{1} = 0 at the point x′. Since the multiplicity of f_{1} is zero at points where f_{m}(1, y_{1}) does not vanish, we see that if we look at all the points x′ in the blown up surface Z which are mapped to our origin by the projection Z → C^{2}, which we denote by x′ → 0, we have

∑_{x′→0} m_{x′}(f_{1}) ≤ ∑_{x′→0} e_{x′} = m ,

Complex curve singularities 33

so that in particular, if there is a point x′ of the strict transform X′ of X which is mapped to 0 and is of multiplicity m on f_{1} = 0, then it is the only point of X′ mapped to 0 and X′ is transversal to the exceptionnal divisor at x′. This fact and its generalizations play a crucial role in Hironaka’s proof of the resolution of singularities.

In order to show that the situation which we have just described cannot persist indefinitely in a sequence of blowing ups, we have to use the intersection number in another manner, according to Hironaka:

Given a germ of a plane curve (X, x) with r branches (X_{i}, x)_{1≤i≤r} and a nonsingular curve W through the point x, define the contact exponent of W with X at x as follows:

δ_{x}(W, X) = min_{i=1}^{r}(\frac{(X_{i}, W )_{x}}{m_{x}(X_{i})} )

and the contact exponent of X at x as follows^{b}

δ_{x}(X) = max_{W} δ_{x}(W, X),

where W runs through the set of germs at x of non-singular curves.

Lemma 6.1. Let f(x, y) = 0 be an equation for X. If the coordinates (x, y) are chosen in such a way that x = 0 is not tangent to X at x and W is defined by y = 0, the rational number δ_{x}(W, X) is the inclination of the first side of the Newton polygon of f(x, y).

By definition of δ_{x}(W, X) is enough to prove that for an irreducible f,

the inclination of the only side of it Newton polygon is ^{\frac{(X,W)_{x}}{m_{x}(X)}} , but if we parametrize X by x = t^{m}, y = t^{q} + ⋯, we find that the transversality condition implies m ≤ q, and we have (X, W )_{x} = q; the result follows.

Lemma 6.2. Assume that W is the curve y = 0 and that f(x, y) is in Weierstrass form, i.e.,

f(x, y) = y^{n} + a_{1}(x)y^{n−1} + ⋯ + a_{n}(x) a_{i}(x) ∈ C{x},

then the inclination of the first side of its Newton polygon is

δ_{x}(W, X) = min_{1≤i≤n−1} \frac{ν_{0}(a_{i})}{i}.

bThis δ_{x}(X) is not related to the δ invariant of a singularity which measures the diminution of genus due to the presence of the singularity, and which is denoted in the same way. These are the classical notations, however.

34 Bernard Teissier

Here as usual ν_{0}(a(x)) denotes the order of vanishing at the origin of the series a(x).

Indeed, the point (0, n) is a vertex of the first side of the Newton polygon, and the lemma is just the observation that if we write a_{i}(x) = α_{i}x^{c_{i}} + ⋯, the other vertices of the Newton polygon are among the points (c_{i}, n − i), which follows directly from the definition.

A nonsingular curve W such that δ_{x}(W, X) = δ_{x}(X) is said to have maximal contact at x. non singular curves with maximal contact are the nonsingular curves which it is hardest to separate from X by a succession of blowing ups (in the sense of separating strict transforms), and so when they eventually separate, something nice should happen; indeed once they separate, there is no point of multiplicity m_{x}(X) in the iterated strict transform mapping to x. As one says, ”the multiplicity has dropped”. Hironaka’s approach to resolution uses the existence of varieties with maximal contact to build an induction on the dimension.

The next step is to prove the existence of curves with maximal contact.

Assume that a non singular curve W defined by y = 0 does not have maximal contact with X at x. We way assume that the curve x = 0 is transversal to f(x, y) = 0, which means that f(0, y) = a_{0,m}y^{m} + ⋯, where m is the multiplicity of f at 0. By a change of variable y = (a_{0,m})^{\frac{1}{m}} y′, which does not change the contacts, we may assume that a_{0,m} = 1. To say that δ_{x}(W, X) < δ_{x}(X) means that there is a series A(x) such that the contact of the curve f(x, y) = 0 with y − A(x) = 0 is greater than its contact with y = 0. By a change of the coordinate x which does not affect the contacts, we may assume that A(x) = ξx^{d} for some integer d and ξ ∈ C^{∗}. Let us now compute the power series expansion in the coordinates x′ = x, y′ = y − A(x);

f′(x, y′) = ∑_{\frac{i}{δ} +j≥m} a_{i,j}x^{i}(y′ + ξx^{d})^{j} = ∑_{\frac{k}{δ′} +ℓ≥m} a′_{k,ℓ}x^{k}y′^{ℓ}.

By expanding the powers of y′ + ξx^{d} we get, for each (i, j), and k ≤ j the inequality \frac{i+kd}{δ} + j − k ≥ m but we know that \frac{i}{δ} + j ≥ m. From this follows the inequality d ≥ δ. Isolating the terms which lie on the first side of the Newton polygon, we get:

(∗) ∑_{\frac{i}{δ} +j=m} a_{i,j}x^{i}y^{j} + ∑_{\frac{i}{δ} +j>m} a_{i,j}x^{i}y^{j} = ∑_{\frac{k}{δ′} +ℓ≥m} a′_{k,ℓ}x^{k}y′^{ℓ},

Complex curve singularities 35

and the slope of the first side of the Newton polygon of the right-hand side is δ′ > δ. Let us first assume that δ = 1. Remark that all the terms x^{k}y′^{ℓ} with \frac{k}{δ′} + ℓ ≥ m except y′^{m} are in the ideal (x, y′)^{m+1}. Therefore we must have the equality

∑_{\frac{i}{δ} +j=m} a_{i,j}x^{i}y^{j} = y′^{m} mod.(x, y)^{m+1}

so that the left hand side is the m-th power of y − ξx^{d}. This implies that d = 1 = δ since the left hand side is homogeneous.

If δ > 1 we follow the same method. Since we know that d ≥ δ, it is easy to check that the ideal of k[[x, y]] generated by the monomials x^{k}y′^{ℓ} , \frac{k}{δ′} + ℓ ≥ m , k ≠ 0 is contained in the ideal I generated by the monomials x^{i}y^{j} , \frac{i}{δ} + j > m. Looking at the equation (∗) modulo I gives us

∑_{\frac{i}{δ} +j=m} a_{i,j}x^{i}y^{j} = y′^{m} mod.I

which again by homogeneity shows that d = δ and the sum on the left hand side is (y − ξx^{d})^{m}.

Note that this argument also works if δ_{x}(X) = ∞. So there are two possibilities:

1) We have δ_{x}(W, X) < δ_{x}(X); in this case the sum of the terms of f(x, y) lying on the first side of the Newton polygon is of the form (y − ξx^{d})^{m}.

2) The sum of the terms of f(x, y) lying on the first side of the Newton polygon is not of the form (y − ξx^{d})^{m}. In the first case, as we have seen, d = δ_{x}(W, X). We make the change of variables x′ = x; y′ = y −ξx^{d} and in the new coordinates x′, y′, if W′ is the curve y′ = 0, we have δ_{x}(W′, X) > δ_{x}(W, X). This follows easily from the computation we have just made; an effect of the change of variables is that all the terms lying on the first side of the Newton polygon, of inclination d, are transformed into the single term monomial y′^{m}. So the inclination of the new Newton polygon has to be > d; but we know this inclination to be δ_{x}(W′, X). If we have not reached δ_{x}(X), we continue the same procedure, and after possibly infinitely many steps, i.e., after a change of variables of the form

x′ = x ; y′ = y − ξ_{1}x^{d_{1}} − ξ_{2}x^{d_{2}} − ⋯ − ξ_{r}x^{d_{r}} − ⋯

we reach the stage where the sum of terms on the first side of the Newton polygon is not a m − th power, so δ_{x}(W_{s}, X) = δ_{x}(X), with s possibly infinite. Since the denominators of the δ_{x}(W, X)’s are bounded, the series

36 Bernard Teissier

is infinite only in the case where δ_{x}(X) = ∞. At least formally this series converges, since we have d_{1} > d_{2} > ⋯ > d_{r} > …, but we can omit the proof of convergence if we work in C{x, y} since the equality δ_{x}(X) = ∞ means that in some coordinates f(x, y) is of the form u(x, y)y^{m} where u is an invertible element in k[[x, y]]; indeed for any other case, we see from the definition that δ_{x}(X) < ∞. But the Weierstrass preparation theorem tells us that if such a presentation exists with formal power series, it also exists with convergent power series, so that the series defining our final coordinates converges.

So in all cases, we can find a nonsingular curve W which has maximal contact with X at x, i.e., such that δ_{x}(W, X) = δ_{x}(X). Remark that all the discussion above is valid on a germ of a non singular surface, since it is analytically isomorphic to the plane. The definition of the blowing up is independant of the choice of coordinates, and makes sense on any nonsingular surface.

The next step is to study the behavior of the contact under blowing up of the origin. I will leave the proof of this as an exercise, since it is a direct application of what we have just seen and the definition of blowing up:

Theorem 6.1. (Hironaka) Let m be an integer, let f(x, y) = 0 define a germ of a plane curve, (X, 0) ⊂ (C, 0) of multiplicity m and let (W, 0) ⊂ (C, 0) be a non singular curve with maximal contact with X at 0. If, after blowing up the point 0 by the map B_{0} : Z → C^{2}, there is a point x′ ∈ X′ of multiplicity m in the strict transform X′ ⊂ Z of X, then

1) The point x′ is the only point of X′ mapped to 0 by B_{0},

2) The strict transform W′ of W by B_{0} contains the point x′, and W′ has maximal contact with X′ at x′,

3) We have the equality δ_{x′}(W′, X′) = δ_{x}(W, X) − 1.

Corollary 6.1. The maximal length of a sequence of infinitely near points of multiplicity m on the strict transforms of X, each mapping to its predecessor in successive blowing ups

⋯ → Z^{(r)} → Z^{(r−1)} → ⋯ → Z^{(2)} → Z^{(1)} → C^{2}

is equal to the integral part [δ_{x}(X)].

This suffices to show that unless the curve is of the form y^{m} = 0, the multiplicity of its strict transform in the sequence of blowing ups obtained by blowing up at each step the points of maximal multiplicity drops after a finite number of steps. By induction on the multiplicity, this proves the

Complex curve singularities 37

resolution of the singularity of X at 0 by a finite number of blowing ups of points on non singular surfaces.We should remark that the map X′ → X

of the strict transform of X to X is defined by itself, without any reference to an embedding (X, 0) ⊂ (C^{2}, 0) (see [K]).

We have proved a local result, but if now we consider any algebraic or analytic curve, it has finitely many singular points, and the local resolution processes at each point are independant, so we have:

Theorem 6.2. Given an algebraic or analytic plane curve X there exists a finite sequence of point blowing ups such that in the composed map X′ → X the curve X′ has no singularities.

Actually we can get, by the same method, a better result, known as embedded resolution and originally due to Max Nœther, as follows:

Theorem 6.3. Given a curve X on a non singular surface S, there exists a finite sequence of blowing ups of points

S^{(r)} → ⋯ → S^{(1)} → S

such that if we denote by π: S^{(r)} = S′ → S their compositum, then the inverse image of the singular points of X (the exceptionnal divisor) is a union of non singular curves ( each isomorphic to P^{1}(C)) meeting transversally on the non singular surface S′, and the strict transform X′ of X by π is a non singular curve meeting transversally these curves.

In analytic terms, if f(x, y) = 0 is a local equation for X in S, then f ∘ π is, at every point x′ of S′, of the form (f ∘ π)_{x′} = u^{a}v^{b} for suitable local coordinates of S′ at x′. Of course a and b will be zero unless we have x′ ∈ π^{−1}(X).The induced map π: X′ → X is a resolution of singularities of X. If we fix a singular point x ∈ X, let r be the number of analytically irreducible components of the germ (X, x). The number of points in π^{−1}(x) is equal to r and for a small enough representative X_{x} of the germ (X, x), the part π^{−1}(X_{x}) of X′ lying over X_{x} consists of r non singular curves D_{i}, each marked with one of the points of π^{−1}(x). The image by π of each of these non singular curves D_{i} is one of the irreducible components of X_{x}.

If we choose for each non singular curve D_{i} a coordinate t_{i} vanishing at the only point z_{i} of D_{i} lying over x, then D_{i} is described parametrically, in local coordinates (u, v) on S′ centered at z_{i}, by convergent power series u(t_{i}), v(t_{i}), because of the implicit function theorem. Since the map π: S′ → S is a composition of algebraic maps, x∘π and y ∘π are at worst convergent

38 Bernard Teissier

power series in (u, v), so when we restrict them to D_{i}, we get convergent power series in t_{i}. This shows that each branch of our curve has a convergent parametrization, and from this we deduce that the formal parametrization constructed by Newton’s method converges.

Note that this convergence argument works equally well with the first resolution theorem. The new fact in the resolution result above with respect to the resolution theorem is the transversality of the strict transform with the exceptionnal divisor, which is not part of the resolution theorem as we have stated it above. The proof of this improvement is not difficult: it amounts to resolving singularities, by a sequence of points blowing up, of the union of the strict transform and the exceptional divisor of the map which resolves the singularities of X. As an example, given an integer m > 1, after one blowing up the strict transform of a curve with equation y^{m} − x^{m+1} = 0, is non singular, but it is not transversal to the exceptionnal divisor.

It is the first example of a fundamental fact of analytic or algebraic geometry: you can make spaces (in fact, their strict transforms) transversal by well chosen sequences of blowing ups.

7. Resolution of space curves

7.1. Integral dependance

To prove a resolution theorem for space curves, one meets the difficulty that their equations may be complicated (for example to define a curve in C^{n} one may need more that n − 1 equations; those for which n − 1 equations suffice are called complete intersections, and also that rather different looking sets of equations may generate the same ideal in C{x_{1}, …, x_{n}} and therefore define the same curve. In the proofs above we have used constructions which depend heavily on the equation. Moreover, even to show that a germ of a complex curve in C^{d} has a finite number of irreducible components, which are analytic germs, is not completely trivial (see [L ], II.5). There are two possibilities: we can conceptualize and abstract the proof for plane curves to make it less dependent on the equation, or try to reduce to the plane curve case. As it happens, the two methods are not so different, at least for one of the ways of abstracting the ideas. To reduce to the plane curve case, the natural idea is to project the space curve X to a plane curve X_{1}. One can then show that a resolution of X_{1} has to map to X, and that this map is a resolution of the singularities of X!.

Complex curve singularities 39

The key idea is that of normalization. The Italian geometers called normal a projective variety Z ∈ P^{n} having the property that any map Z′ → Z presenting Z as a ”general” projection by a linear map P^{n′} \ L → P^{n} of an algebraic variety Z′ ⊂ P^{n′} had to be an isomorphism. A typical non normal surface in P^{3} is therefore a general projection of a non singular surface in P^{4} which cannot be embedded in P^{3}; such a projection has a curve of double points, on which are finitely many more complicated singular points, the ”pinch points”. Here the meaning of ”general” has to be made precise; The variety Z is normal if any map π: Z′ → Z which

a) is finite-to-one and

b) induces an isomorphisme Z′ \ π^{−1}(U) → U, over the complement U of a closed algebraic or analytic subset of Z of smaller dimension, is an isomorphism.

The resolution theorem we saw above shows that a singular curve in P^{2} cannot be normal.

The concept of normalization was ”localized” and transfigured into a concept of commutative algebra as follows: Recall that the total ring of quotients of a ring A is the ring of equivalence classes of couples (a, b) of elements of A, where b is not a zero divisor in A with addition (a, b) + (a′, b′) = (ab′ + ba′, bb′) and component wise multiplication, the equivalence being (a, b) ≡ (a′, b′) when ab′ − ba′ = 0. The map a ↦ (a, 1) induces an injection of A in F and we indentify A with its image in F. If A is an integral domain, F is its field of fractions.

Definition Let A be a commutative ring without nilpotent elements, and let F be its total ring of quotients. Definition.- An element h ∈ F is integral over A if it satisfies an equation

h^{k} + a_{1}h^{k−1} + ⋯ + a_{k} = 0 with a_{i} ∈ A.

Example.- Consider the germ of plane curve X in C^{2} defined by the equa-

tion y^{p} − x^{q} = 0. The quotient O of the ring C{x, y} by the ideal generated by y^{p} − x^{q} is the ring of germs of analytic functions on the germ X (the restrictions to X of two analytic functions on C^{2} coincide if and only if their difference is in the ideal). The ring O is an integral domain; let K be its field of fractions. If we keep the notations x, y, etc.. for the restrictions to X of functions on C^{2}, we have \frac{y}{x} ∈ F. I claim that if p ≤ q, it is integral over O; indeed, we have the relation

(\frac{y}{x})^{p} − x^{q−p} = 0 .

40 Bernard Teissier

We can remark that the function \frac{y}{x} is defined and analytic on the strict transform of X by the blowing up of the origin for any sufficiently small representative of the germ X. We remark also that the condition p ≤ q is equivalent to saying that the meromorphic function \frac{y}{x} remains bounded on X for any small representative.

Proposition 7.1. Given a ring A without nilpotent elements, let F be its total ring of fractions; the set of elements of F integral over A is a ring for the operations induced by those of F.

This ring is called the normalization of A (or the integral closure of A in F) and often denoted by \overline{A}. Of course we have A ⊂ \overline{A}; a ring such that A = \overline{A} is said to be integrally closed. Is is not difficult to check that \overline{A} is integrally closed.

If A is nœtherian and integrally closed, any injective map A → B to a subring B of the total ring of fractions of A which makes B into a finite Amodule is an isomorphism; this is the translation of the original definition of normality. To prove it, check that if h is an element of B, the powers of h cannot all be linearly independant over A, so h satisfies an integral dependance relation, and if A is normal, it is in A!

An important theorem is that if A is an analytic algebra, i.e., a quotient of a convergent power series ring by some ideal, then \overline{A} is a finite sum of integrally closed analytic algebras, and moreover that the injection A → \overline{A} makes \overline{A} into a finitely generated A-module. Taking a common denominator (in F) for a finite set of generators of the A-module \overline{A}, we see that the (”conductor”) ideal C = \{d ∈ \overline{A}, d.\overline{A} ⊂ A\} is not zero. Another important fact is that if the analytic algebra of germs of functions on a curve at a point is normal, the point is non singular on the curve, and the analytic algebra is isomorphic to a convergent power series ring in one variable C{t}. ([L ], VI.3, Thm.2)

7.2. The δ invariant of a plane curve singularity

Let O be the analytic algebra of a germ of curve (X, 0), plane or not, and let \overline{O} be its normalization. Since it is an O-module of finite type with the same total ring of quotients, a version of the Hilbert Nullstellensatz shows that the quotient vector space over C is finite dimensional. So we may define an invariant to measure how far O is from being integrally closed, i.e., regular:

δ_{X,0} = dim_{C} \frac{\overline{O}}{O}

Complex curve singularities 41

In the case of plane curves, this invariant is known as the diminution of genus which the presence of the singularity imposes on the curve: Take a projective plane curve C of degree d. If it is non singular, it is topologically a differentiable surface, classified by its genus. This genus is equal to

g(C) = \frac{(d − 1)(d − 2)}{2}.

If the curve has singularities, then its normalization is topologically a differentiable surface, and its genus is

g(\overline{C}) = \frac{(d − 1)(d − 2)}{2} − Σ_{x∈C}δ_{C,x},

where the sum on the right is finite since δ_{C,x} is nonzero only at singular points, and those are finite in number. The genus of the normalization of C is traditionally called the geometric genus of C. Moreover, the local invariant δ_{C,x} has a local geometrical interpretation, (see [T4], [T6]) which I will describe only in the case of a branch, for simplicity: Let t^{n}, y(t) be a parametrization of our branch X. Consider the product of the normalization of X with itself, with coordinates (t′t′) and the two curves in (C^{2}, 0) = (\overline{X} × \overline{X}, 0) defined by

\frac{t^{n} − t′^{n}}{t − t′} = 0 \frac{y(t) − y(t′)}{t − t′} = 0

The intersection number of these two curves at the origin is equal to 2δ_{X}; if now we perturb slightly the parametrization of X by t^{n} + αvt, y(t) + βvt with two ”general ” complex numbers α, β, we can see that the two curves now have deformed equations and for small v they now meet transversally in 2δ_{X} points in \overline{X} ×\overline{X}. this means that the curve defined parametrically by t^{n} + vαt, y(t) + vβt has δ_{X} ordinary double points (two branches meeting transversally), which tend to 0 as v tends to 0. So we can view δ_{X} as the number of ordinary double points which have coalesced to form the singularity of X at the origin. Of course, for an ordinary double point δ = 1.

In fact this geometric interpretation follows from the fact that the δ invariant plays a key role in understanding which deformations of curves come from deforming the parametrization.

42 Bernard Teissier

If a germ of plane curve is given parametrically by x(t), y(t), we can define (one parameter) deformations of the parametrization as follows:

\begin{aligned}x(t; v) & = x(t) + ∑_{i} a_{i}(v)t^{i}, a_{i} ∈ C\{v\}, a_{i}(0) = 0 \\ y(t; v) & = y(t) + ∑_{j} b_{j}(v)t^{j}, b_{j} ∈ C\{v\}, b_{j}(0) = 0. \end{aligned}

If on the other hand our curve is given implicitely by an equation f(x, y) = 0, then we can define a deformation as

f(x, y; v) = f(x, y) + ∑_{(i,j)≠(0,0)} g_{ij}(v)x^{i}y^{j} g_{ij} ∈ C{v}, g_{ij}(0) = 0.

The elimination process can be performed over C{v} to show that a deformation of the parametrization always give a deformation of the equation (again this follows from the fact that the formation of Fitting ideals commutes with base extension). Is the converse true in the sense that any deformation of the equation can be represented by a deformation of the parametrization? The answer is NO!

In order to understand what happens, we must reinterpret the problem. To say that a family of curves is obtained by deforming a parametrization is to say that they all have “the same normalization” in some sense. Thus we are led to study how the normalizations vary in an analytic family of reduced plane curves.

Definition 7.1. Let (C, 0) be a germ of a reduced analytic curve, or let C be a closed reduced analytic curve in a suitable open polycylinder of C^{d}, or an affine curve, with ring O_{C}. Then its δ invariant is defined by δ_{0}(C) = dim_{C}(\frac{\overline{O_{C,0}}}{O_{C,0}} ), or δ(C) = dim_{C}(\frac{\overline{O_{C}}}{O_{C}} ).

Since normalization of a sheaf of algebras form a sheaf, we have

δ(C) = ∑_{x∈C} δ_{x}(C)

where the sum on the right is finite since δ is nonzero only at singular points.

Now let f: (S, 0) → (C, 0) be a germ of a flat morphism such that f^{−1}(0) is a germ of a reduced analytic curve. Here flatness means that no element of O_{S,0)} is annihilated by multiplication by an element of C{v} where v is a local coordinate on (C, 0). Let n: \overline{S} → S be the normalization of the surface S (a small representative of the germ), and let

p = f ∘ n: (\overline{S}, n^{−1}(0)) → (S, 0).

Complex curve singularities 43

Let us denote p^{−1}(0) by (\overline{S})_{0}, and to write δ((\overline{S})_{0}) = ∑_{x∈n^{−1}(0)} δ((\overline{S})_{0}, x). similarly, write δ(S_{0}) for δ(f^{−1}(0), 0) and δ(S_{y}) for δ(f^{−1}(y) when y ∈ C \ {0} in a small enbough representative of f, so that all the singular points of f^{−1}(y) tend to 0 when y → 0, and 0 is the only singular point of f^{−1}(0). Note that δ(S_{y}) = ∑_{z∈S_{y}} δ(S_{y}, z). Then we have:

Proposition 7.2. (see [T6], and [CH-L] for a beautiful generalization)

a) The morphism p = f ∘ n: (\overline{S}, n^{−1}(0)) → (C, 0) is a multigerm of a flat mapping.

b) We have the equality

δ((\overline{S})_{0}) = δ(S_{0}) − δ(S_{y}),

for y ≠ 0 sufficiently small.

To say that the normalizations of the various fibers f^{−1}(y) glue up into a non singular surface is therefore equivalent to saying that p^{−1}(0) is non singular and this is equivalent to saying that “the δ invariant of the fibers S_{y} is constant as y varies in C near 0.

Note that the fiber f^{−1}(y) will in general have several singular points, at which it is not necessarily analytically irreducible even if f^{−1}(0) is irreducible.

This explains what happens when we deform the parametrization by x(t)+ αvt, y(t) + βvt; since it is a deformation of the parametrization, the sum of the δ invariants must be the same for all values of v, while for v ≠ 0 the curve has only ordinary double points, whose δ invariant is one.

7.3. projections of space curves

So this abstract idea, normalization, provides us with a proof of the resolution of singularities of space curves: given (C, 0) ∈ (C^{d}, 0), the normalization O → \overline{O} of the (reduced) analytic algebra of germs of functions on C is an analytic algebra which is a product Π_{i=1}^{r}C{t_{i}} of a finite number of convergent power series rings in one variable. If x_{1}, …, x_{d} generate the maximal ideal of O, we get r d-uples of convergent power series expansions x_{j}(t_{i}), which are our Newton series in this case. They geometrically correspond to a map

⨆_{i=1}^{r}(C, 0)_{i} → (C, 0)

44 Bernard Teissier

which is our resolution of singularities. However, normalization is geometrically subtle in general, and the finiteness of normalization for the rings one meets in Geometry is a fairly deep theorem of commutative algebra; in addition, we may seek a more geometric proof, as follows

We now turn to the definition of plane projections of a space curve. Let (C, 0) ∈ (C^{d}, 0) be a germ of a (reduced) space curve defined by an ideal I ⊂ C{x_{1}, …, x_{d}}. Let us choose a linear projection p: C^{d} → C^{2}. Let M denote the space of all such projections; think of it as a set of d×2 matrices of rank 2. We endow M with the topology (complex or Zariski) induced by that of the space of matrices. We wish to consider only the projections such that p|C: C → p(C) is finite to one. If that is not the case, the kernel of p, which is a linear subspace of codimension 2 of C^{d}, contains one of the irreducible components of the curve C; the intersection is analytic, so it is either of dimension 0 or 1. By looking at the equations of C, it is not too difficult to check that the projections which do not contain a component of C form a dense open set of M. The fact that they are those which induce a finite map C → p(C) is a consequence of the Weierstrass preparation theorem. Assume now that the map C → p(C) is finite. Again by the Weiertrass theorem, it means that the map of analytic algebras C{x, y} → O defined by f ↦ (f ∘ p)|C makes O a C{x, y}-module of finite type. Since C{x, y} is nœtherian, as we saw in a preceding section, it means we have a presentation by an exact sequence of C{x, y}-modules:

C{x, y}^{q} → C{x, y}^{p} → O → 0

An argument which we have seen above shows that since C is of dimension 1, we must have q = p, so the first map is described by a square matrix with entries in C{x, y}. Let ϕ(x, y) be the determinant of that matrix. This determinant is, up to an invertible factor, independant of the choice of the presentation. Then the image p(C) is the plane curve with equation ϕ(x, y) = 0.

On the other hand, let us say that a linear plane projection p: C^{d} → C^{2} is general for the curve C ⊂ C^{d} at the point 0 ∈ C if it has the following property: For any sequence of couples of points (a_{i}, b_{i}) ∈ (C \{0})×(C \{0}) tending to 0, the limit direction of the secant line \overline{a_{i}, b_{i}} (for any subsequence) is not contained in the kernel of p.

We will see in the next paragraph that all general projections of a given

Complex curve singularities 45

germ (C, 0) of space curve are topologically indistinguishable as germs of plane curves in C^{2}. In [T2] it is shown that if p is general for (C, 0), then the inclusion of the ring O_{1} = O_{X_{1},0} of the image X_{1} = p(X) as defined above into the ring O = O_{C,0} (induced by the composition of functions with p) induces an isomorphism of the total rings of fractions of these two rings, and because O is a finite O_{1}-module, every element of O is integral over O_{1}, as we saw above. Therefore O is contained in the normalization \overline{O_{1}} of O_{1}. Therefore \overline{O_{1}} is also the normalization of O, and it is a finite O-module for general reasons (see [K]; it suffices to know that the integral closure of O_{1} is a finite O_{1}-module). Now we can use the universal property of blowing ups: in \overline{O_{1}} all ideals become principal and generated by a non zero divisor in each C{t_{i}}. By the universal property of blowing up ([L ], VII.5) if we blow up the origin in O, the resulting algebra is still contained in \overline{O_{1}}, and as we repeat blowing up points, we get an increasing sequence of O-algebras contained in \overline{O_{1}}, all having the same total ring of fractions. Since \overline{O_{1}} is a finite O-module, this sequence stabilizes after finitely many steps. We have to show that this limit algebra is \overline{O_{1}}. But if this were not the case, the maximal ideal of one of the component local algebras would not be principal, so we could blow it up and get a strictly bigger algebra, contradicting the stability. In conclusion, we have shown that any space curve singularity can also be desingularized by a finite sequence of point blowing ups. One can also prove embedded resolution for space curves; it is not much more difficult than in the plane curve case.

8. The semigroup of a branch

There is another natural object associated to the inclusion O → \overline{O}; again I will decribe it only in the case of a branch.

Let O be the analytic algebra of a germ of analytically irreducible curve X, and let \overline{O} be its normalization; we have an injection O → \overline{O} which makes \overline{O} an O-module of finite type and \overline{O} is a subalgebra of the fraction field of O. Since \overline{O} is isomorphic to C\{t\}, the order in t of the series defines a mapping ν: C{t} \ 0 → N which satisfies

i) ν(a(t)b(t)) = ν(a(t)) + ν(b(t)) and ii) ν(a(t) + b(t)) ≥ min(ν(a(t)), ν(b(t))) with equality if ν(a(t)) ≠ ν(b(t)); in other words, ν is a valuation of the ring C{t}.

We consider the valuations of the elements of the subring O, i.e., the

46 Bernard Teissier

image Γ of O \ {0} by ν; in view of i), it is a semigroup contained in N. The fact that \overline{O} is a finite O-module implies that N \ Γ is finite, and in fact (see [Z]) we have for the δ invariant of C the equality

δ_{X} = #(N \ Γ)

Now we seek a minimal set of generators of Γ as a semigroup: Let \overline{β_{0}} be the smallest non zero element in Γ, let \overline{β_{1}} be the smallest element of Γ which is not a multiple of \overline{β_{0}}, let \overline{β_{2}} be the smallest element of Γ which is not a combination with non negative integral coefficients of \overline{β_{0}} and \overline{β_{1}}, i.e., is not in the semigroup ⟨\overline{β_{0}}, \overline{β_{1}}⟩, and so on. Finally, since N\Γ is finite, we find in this way a minimal set of generators:

Γ = ⟨\overline{β_{0}}, \overline{β_{1}}, …, \overline{β_{g}}⟩

This set is uniquely determined by the semigroup Γ, and of course determines it. By a theorem of Apéry and Zariski (see [Z]), if (X, 0) is a plane branch, the datum of these generators, or of the semigroup, is equivalent to the datum of the Puiseux characteristic of (X, 0), or of its topological type.

Let us take the notations introduced for the Puiseux pairs; it is easy to check that if we set β_{0} = n, the multiplicity, then \overline{β_{0}} = β_{0} = n, \overline{β_{1}} = β_{1}. After that it becomes more complicated. Zariski ([Z], Th. 3.9) proved the following formula for q = 2, …, g:

\overline{β_{q}} = (n_{1} − 1)n_{2} … n_{q−1}β_{1} + (n_{2} − 1)n_{3} … n_{q−1}β_{3} + ⋯ + (n_{q−1} − 1)β_{q−1} + β_{q},

which can be summarized in the following recursive formula:

\overline{β_{q}} = n_{q−1}\overline{β_{q−1}} − β_{q−1} + β_{q}

The proof relies on a formula of Max Noether which computes the contact exponent \frac{(C,D)_{0}}{m_{0}(D)} of two analytic branches at the origin in terms of the coincidence of their Puiseux expansions in fractional powers of x. This fact leads to a very interesting constatation: Consider the Puiseux expansion of a root y(x) of the Weierstass polynomial defining an analytically irreducible plane curve near the origin, assuming that x = 0 is not in the tangent cone of that curve:

y = a_{n}x + a_{2n}x^{2} + ⋯ + a_{kn}x^{k} + a_{β_{1}}x^{\frac{m_{1}}{n_{1}}} + a_{β_{1}+e_{1}}x^{\frac{m_{1}+1}{n_{1}}} + ⋯ + a_{β_{1}+k_{1}e_{1}}x^{\frac{m_{1}+k_{1}}{n_{1}}}

+a_{β_{2}}x^{\frac{m_{2}}{n_{1}n_{2}}} + a_{β_{2}+e_{2}}x^{\frac{m_{2}+1}{n_{1}n_{2}}} + ⋯ + a_{β_{q}} x^{\frac{mq}{n_{1}n_{2}···nq}} + a_{β_{q}+e_{q−1}}x^{\frac{mq +1}{n_{1}n_{2}···nq}} + ⋯

+a_{β_{g}} x^{\frac{mg}{n_{1}n_{2}···ng}} + a_{β_{g}+1}x^{\frac{mg +1}{n_{1}n_{2}···ng}} + ⋯

Complex curve singularities 47

and the following series

ξ_{0} = x,
ξ_{1} = a_{n}x + a_{2n}x^{2} + ⋯ + a_{kn}x^{k}
ξ_{2} = a_{n}x + a_{2n}x^{2} + ⋯ + a_{kn}x^{k} + a_{β_{1}}x^{\frac{m_{1}}{n_{1}}} + a_{β_{1}+e_{1}}x^{\frac{m_{1}+1}{n_{1}}} + ⋯ + a_{β_{1}+k_{1}e_{1}}x^{\frac{m_{1}+k_{1}}{n_{1}}}

⋮

ξ_{g} = y − (a_{β_{g}} x^{\frac{mg}{n_{1}n_{2}···ng}} + a_{β_{g}+1}x^{\frac{mg +1}{n_{1}n_{2}···ng}} + ⋯)

That is, the sequence of truncations of the Puiseux series just before the appearance of a new Puiseux exponent. Each ξ_{j}, 0 ≤ j ≤ g is a root of a Weierstrass polynomial Q_{j} defining a branch C_{j}. Note that we have Q_{0} = x and that Q_{1} = y if y = 0 has maximal contact with C.

Proposition 8.1. (Apéry-Zariski), see also [PP] For the semigroup Γ = ⟨\overline{β}_{0}, \overline{β}_{1}, …, \overline{β}_{g}⟩ associated to a plane branch and the curves C_{j} just defined, we have the equalities

\overline{β}_{j} = (C, C_{j})_{0}.

In fact, this equality remains true if we replace the expansions ξ_{j} by any series which coincide with the series y(x) until just before the j-th Puiseux exponent; see [PP]. It follows easily from this that the datum of the semigroup is equivalent to the datum of the multiplicity n and the Puiseux exponents β_{i} of the curve.

The semigroups coming from plane branches are characterized among all semigroups of analytically irreducible germs of curves by the following two properties:

1) n_{i}\overline{β_{i}} ∈ ⟨\overline{β_{0}}, …, \overline{β_{i−1}}⟩

2) n_{i}\overline{β_{i}} < \overline{β}_{i+1}

That the semigroups of plane branches have these properties follows from the induction formula and the inequalities β_{i} < β_{i+1}. The converse can be proved by the construction outlined below (see [Z], appendix).

Conversely, given a semigroup Γ in N with finite complement, we can associate to it an analytic (in fact algebraic) curve, called the monomial curve associated to Γ. If Γ = ⟨\overline{β_{0}}, \overline{β_{1}}, …, \overline{β_{g}}⟩, the monomial curve C^{Γ} is

48 Bernard Teissier

described parametrically by

u_{0} = t\overline{β_{0}}
u_{1} = t\overline{β_{1}}
.
.
.
u_{g} = t\overline{β_{g}}

If the semigroup Γ comes from a plane branch, the relations 1) above mean that there exist natural numbers ℓ_{i}^{(j)} such that we have

n_{1}\overline{β_{1}} = ℓ_{0}^{(1)}\overline{β_{0}}
n_{2}\overline{β_{2}} = ℓ_{0}^{(2)}\overline{β_{0}} + ℓ_{1}^{(2)}\overline{β_{1}}
.
.
n_{j}\overline{β_{j}} = ℓ_{0}^{(j)}\overline{β_{0}} + ⋯ + ℓ_{j−1}^{(j)}\overline{β_{j−1}}
.
.
.
n_{g}\overline{β_{g}} = ℓ_{0}^{(g)}\overline{β_{0}} + ⋯ + ℓ_{g−1}^{(g)}\overline{β_{g−1}}

These relations translate into equations for the curve C^{Γ} ⊂ C^{g+1}; since u_{i} = t\overline{β_{i}}, our curve satisfies the g equations

u_{j}^{n_{i}} − u_{0}^{ℓ_{0}^{(j)}}u_{1}^{ℓ_{1}^{(j)}} … u_{j−1}^{ℓ_{j−1}^{(j)}} = 0, 1 ≤ j ≤ g,

and it can be shown that they actually define C^{Γ} ⊂ C^{g+1}, so that if Γ is the semigroup of a plane branch, C^{Γ} is a complete intersection.

Remark that if we give to u_{i} the weight \overline{β_{i}}, the i-th equation is homogeneous of degree n_{i}\overline{β_{i}}.

The connection between a plane curve X having semigroup Γ and the monomial curve is much more precise and interesting than the formal relation we have just seen; by small deformations of the monomial curve one obtains all the branches with the same semigroup. In fact the best way to understand all branches with semigroup Γ is to consider the not necessarily plane curve C^{Γ} (C^{Γ} is plane if and only if C has only one characteristic exponent).

By definition of Γ, there are elements ξ_{q} ∈ O with ν(ξ_{q}) = \overline{β_{q}}. We can write these elements in C{t} as

ξ_{q} = t\overline{β_{q}} + ∑_{j>\overline{β_{q}}} γ_{q,j}t^{j}.

Complex curve singularities 49

Let us consider the one-parameter family of parametrizations

u_{0} = t^{m}
u_{1} = t\overline{β_{1}} + ∑_{j>\overline{β_{1}}} v^{j−\overline{β_{1}}}γ_{1,j}t^{j}
.
.
u_{g} = t\overline{β_{g}} + ∑_{j>\overline{β_{g}}} v^{j−\overline{β_{g}}}γ_{g,j}t^{j}

The reader can check that for v ≠ 0, the curve thus described is isomorphic to our original curve C. (hint: make the change of parameter t = vt′ and the change of coordinates u_{j} = v^{−\overline{β_{j}}}v′_{j}, and remember the definition of the ξ_{j}). For v = 0, we have the parametric description of the monomial curve.

So we have in fact described a map

C × C → C^{g+1} × C

which induces the identity on the second factors (with coordinate v). The image of this map is a surface, which is the total space of a deformation of the monomial curve, all of its fibers except the one for v = 0 being isomorphic to our plane curve C.

So the monomial curve is a specialization, in this family, of our plane curve. In this specialization the multiplicity and the semigroup remain constant; in a rather precise sense it is an equisingular specialization, or one may say that the plane curve is an equisingular deformation of the monomial curve with the same semigroup.

The same phenomenon can be also observed in the language of equations rather than parametrizations. Let us consider a one parameter family of equations for curves in C^{g+1}, of the form

u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}} − vu_{2} = 0
u_{2}^{n_{2}} − u_{0}^{ℓ_{0}^{(2)}}u_{1}^{ℓ_{1}^{(2)}} − vu_{3} = 0
.
.
u_{g−1}^{n_{g−1}} − u_{0}^{ℓ_{0}^{(g−1)}}u_{1}^{ℓ_{1}^{(g−1)}} … u_{g−2}^{ℓ_{g−2}^{(g−1)}} − vu_{g} = 0
u_{g}^{n_{g}} − u_{0}^{ℓ_{0}^{(g)}}u_{1}^{ℓ_{1}^{(g)}} … u_{g−1}^{ℓ_{g−1}^{(g)}} = 0

For v = o we get the equations of the monomial curve, and for v ≠ 0 we get a curve which has semigroup Γ; this is a general heuristic principle of equisingularity: we have added to each equation of the monomial curve, homogeneous of degree n_{i}\overline{β_{i}}, a perturbation of degree \overline{β_{i+1}} > n_{i}\overline{β_{i}}, and this

50 Bernard Teissier

should not change the equisingularity class (the perturbation is ”small” compared to the equation).

Notice that for each fixed v ≠ 0 the curve described by the above equations is a plane curve: for simplicity take v = 1; then use the first equation to compute u_{2} = u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}}, substitute this in the next equation, and use this to compute u_{3} as a function of u_{0}, u_{1}, and so on. Finally the last equation gives us the equation of a plane curve of the form

(⋯ ((u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}})^{n_{2}} − u_{0}^{ℓ_{0}^{(2)}}u_{1}^{ℓ_{1}^{(2)}})^{n_{3}} − ⋯)^{n_{g}} −u_{0}^{ℓ_{0}^{(g)}}u_{1}^{ℓ_{1}^{(g)}}(u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}})^{ℓ_{2}^{(g)}} ⋯ = 0

The first consequence (see the appendix to [Z]) is that we can produce explicitely the equation of a plane curve with given characteristic exponents: compute the semigroup and its generators, and then write the equation above.

A more important fact is that one can show (see [loc. cit) that any plane curve with a given semigroup appears up to isomorphism as a fiber in a deformation depending on a finite number of parameters: it is a deformation of the monomial curve obtained by adding to the j-th equation a polynomial in the u_{i}’s of order > n_{j}\overline{β_{j}}, and these polynomials can in principle be explicitely computed.

In fact it is shown in [G-T] that we can in this manner produce equations for all branches having the same semigroup (or equisingularity type) up to an analytic isomorphism. The fact that the curve is plane corresponds to the condition that u_{j+1} appears linearly in the deformation of the j-th equation, for 1 ≤ j ≤ g − 1. Finally, all the plane branches with the same semigroup have ”the same” process of resolution of singularities: you have to blow up points according to the same rules, the multiplicities of the strict transforms are the same, and so on. So the resolution of the plane curve described above shows the structure of the resolutions of all the curves with the same semigroup. First you resolve the curve u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}} = 0; when its strict transform is non singular (after a number of blowing ups which depends on the continued fraction expansion of the ratio \frac{ℓ_{0}^{(1)}}{n_{1}} , you take it as a coordinate axis: then you have one parenthesis less in the equation above (the point is that the form of the equation does not change), and you proceed like this. After g such steps the branch is resolved.

There is however another way to use the structure given by the description of our branch as a deformation of the monomial curve to get embedded resolution; it is the subject of the next paragraph.

Complex curve singularities 51

9. Resolution of binomials

Let a^{1}, a^{2} be two integral vectors in the first quadrant of Ř^{2}, and assume that their determinant is ±1. Then they are primitive vectors and they generate the integral lattice Ž^{2} of Ř^{2}. Consider the cone

σ = ⟨a^{1}, a^{2}⟩

of their positive linear combinations. It is a rational convex cone (= a convex cone which is the intersection of finitely many half spaces determined by hyperplanes with rational -even integral- equations). Because it is generated by integral vectors which form a basis of the integral lattice Z^{2}, we say tha it is a regular cone. Since it is convex it has a convex dual which is a rational convex cone in R^{2}:

σ̌ = {m ∈ R^{2}/m(ℓ) ≥ 0 ∀ ℓ ∈ σ}.

The cone σ̌ is also generated by two vectors with determinant ±1, which therefore generate the integral lattice Z^{2} of R^{2}. If we interpret each integral point of σ̌ as a (Laurent) monomial (here “Laurent” means that negative exponents are allowed) in variables (u_{1}, u_{2}), the algebra C[σ̌ ∩ Z^{2}] is a polynomial algebra in two variables, say C[y_{1}, y_{2}].

Since σ is contained in the first quadrant, its dual σ̌ contains the dual of the first quadrant, which is the first quadrant of R^{2}. If we remark that the integral points of the first quadrant correspond exactly to the polynomial algebra C[u_{1}, u_{2}], we see that there is therefore an inclusion

C[u_{1}, u_{2}] ⊂ C[y_{1}, y_{2}]

and it is an interesting exercise to check that it is given by

\begin{matrix}u_{1} & ↦ y_{1}^{a_{1}^{1}}y_{2}^{a_{1}^{2}} \\ u_{2} & ↦ y_{1}^{a_{2}^{1}}y_{2}^{a_{2}^{2}} \end{matrix}

where a_{i}^{j} is the i-th coordinate of the vector a^{j}.

The transform of a monomial u^{m} = u_{1}^{m_{1}}u_{2}^{m_{2}} is, if we write m = (m_{1}, m_{2}):

u_{1}^{m_{1}}u_{2}^{m_{2}} ↦ y_{1}^{⟨a^{1},m⟩}y_{2}^{⟨a^{2},m⟩},

so that the transform of a binomial u^{m} − λ_{mn}u^{n} is

u^{m} − λ_{mn}u^{n} ↦ y_{1}^{⟨a^{1},m⟩}y_{2}^{⟨a^{2},m⟩} − λ_{mn}y_{1}^{⟨a^{1},n⟩}y_{2}^{⟨a^{2},n⟩}.

Now the key observation is that if ⟨a^{1}, m − n⟩ and ⟨a^{2}, m − n⟩ are both non zero, they have the same sign, which means that the two vectors a^{1} and a^{2}

52 Bernard Teissier

are in the same half space determined the hyperplane H_{m−n} dual to the vector m − n, or equivalently that the cone σ is compatible with H_{m−n} in the sense that σ ∩ H_{m−n} is a face of σ, then we can factor the transform of the binomial. Assume that ⟨a^{i}, m−n⟩ ≥ 0. We have non negative exponents in the identity

\begin{matrix}y_{1}^{⟨a^{1},m⟩}y_{2}^{⟨a^{2},m⟩} − λ_{mn}y_{1}^{⟨a^{1},n⟩}y_{2}^{⟨a^{2},n⟩} = \\ y_{1}^{⟨a^{1},n⟩}y_{2}^{⟨a^{2},n⟩}(y_{1}^{⟨a^{1},m−n⟩}y_{2}^{⟨a^{2},m−n⟩} − λ_{mn}). \end{matrix}

Now we have an exceptional divisor defined by

y_{1}^{⟨a^{1},n⟩}y_{2}^{⟨a^{2},n⟩} = 0

and a strict transform defined by

y_{1}^{⟨a^{1},m−n⟩}y_{2}^{⟨a^{2},m−n⟩} − λ_{mn} = 0

The next observation is that the strict transform is non singular, and meets the exceptional divisor if and only if σ ∩ H_{m−n} is an edge of σ, i.e., is not {0}. Say that a^{2} is in H_{m−n}; the strict transform is then

y_{1}^{⟨a^{1},m−n⟩} − λ_{mn} = 0.

Now if we assume that the binomial u^{m} −λ_{mn}u^{n} is irreducible in C[u_{1}, u_{2}], which is equivalent since C is algebraically closed to the fact that the vector m − n is primitive, in the sense that it is not an integral multiple of an integral vector, then it is not difficult to check (see [T1], Proposition 6.2) that ⟨a^{1}, m − n⟩ = 1, so that finally our strict transform in this case is y_{1} − λ_{mn} = 0, which is indeed non singular and transversal to the exceptional divisor. Actually the same proof works if the binomial is reducible but there are then several points above the origin in the strict transform of the curve. The next observation is that in two variables our binomial has to be of the form u_{1}^{m} − λu_{2}^{n} unless the curve contains a coordinate axis, which we exclude in the irreducible case. By a change of variable we may assume λ = 1 and by irreducibility, we have (m, n) = 1. Now to study the strict transform under one of our monomial maps π(σ) we have seen that the only interesting case is when one of the generating vectors of σ, say a^{1}, is the vector (n, m). Let us assume that n < m. Set a^{2} = (a, b) and say that am − bn = 1 (we know it has to be ±1). The transform of a monomial u_{1}^{i}u_{2}^{j} is y_{1}^{ni+mj}y_{2}^{ai+bj}. From this follows that if we consider a curve with equation

(∗)  u_{1}^{m} − u_{2}^{n} + ∑_{ni+mj>mn} a_{ij}u_{1}^{i}u_{2}^{j} = 0

Complex curve singularities 53

it transforms into

y_{1}^{mn}y_{2}^{am}(y_{2} − 1 + ∑_{ni+mj>mn} a_{ij}y_{1}^{ni+mj−mn}y_{2}^{ai+bj−am})

and one checks that all exponents are positive. The strict transform of our curve is still non singular in a neighborhood of the exceptional divisor, and transversal to the exceptional divisor at the point y_{1} = 0, y_{2} = 1. If we consider the other cone σ′ having the vector (n, m) as an edge, we find that the point where the strict transform meets the exceptional divisor lies in the open set of the corresponding chart Z(σ′) which is identified with an open set of Z(σ); we are looking at the same object in two charts. This shows that the toric maps which provides an embedded resolution for the binomial u_{1}^{m} − u_{2}^{n} = 0 in fact also gives an embedded resolution for all the curves of the type (∗), where one deforms the binomial by adding terms of higher weight, where the weight of u_{1} is n and the weight of u_{2} is m. Now by a general combinatorial result (see [Ew]), for any integer d ≥ 2, given a finite collection of hyperplanes whose equation has integral coefficients in the first quadrant R_{≥0}^{d} of R^{d}, it is possible to find a regular fan with support R_{≥0}^{d}, that is a finite collection Σ of regular rational cones such as our σ above (but now with d generating vectors of determinant ±1) and its faces, whose union is R_{≥0}^{d}, and such that if σ ∈ Σ its faces are also in Σ, and for any σ, σ′ ∈ Σ, the intersection σ ∩ σ′ is a face of each. To each σ of dimension d corresponds a polynomial ring C[σ̌ ∩ Z^{d}] and therefore an affine space A^{d}(C) with a birational map

π(σ): A^{d}(C) → A^{d}(C)

generalizing the map

\begin{matrix}u_{1} & ↦ y_{1}^{a_{1}^{1}}y_{2}^{a_{1}^{2}} \\ u_{2} & ↦ y_{1}^{a_{2}^{1}}y_{2}^{a_{2}^{2}} \end{matrix}

which we have seen above in the case d = 2. The sources of all these maps can be glued up together (see [Ew]) to form a nonsingular algebraic rational variety Z(Σ) in such a way that the maps π(σ) glue up into a proper and birational (hence surjective) map

π(Σ): Z(Σ) → A^{d}(C).

Coming back to the case d = 2 and a binomial, this gives us the existence of a regular fan (= a fan made of regular cones) with support R_{≥0}^{2}, and compatible with the line H_{m−n}, which means that this line in R_{≥0}^{2} is the common edge of two cones of the fan.

54 Bernard Teissier

In fact in this case there is a minimal such fan, obtained as follows: Consider the set H_{−} of integral points of R_{≥0}^{2} which are below the line H_{m−n}, and the set H_{+} of the integral points which are above. The boundaries of the convex hulls of H_{−} and H_{+} contain parts of coordinates axes, and they meet at the extremity of the primitive integral vector contained in H_{m−n}. Drawing lines connecting the origin to all the integral points which are on these boundaries defines a fan which has the required properties and is the coarsest such fan. It is closely connected with the continued fraction expansion of the slope of the line H_{m−n}. To this fan Σ is associated a proper birational map

π(Σ): Z(Σ) → A^{2}(C)

which is an isomorphism outside of the origin and provides an embedded resolution of singularities for all plane branches which have an equation of the form

u_{2}^{n} − u_{1}^{m} + ∑_{\frac{i}{m} + \frac{j}{n} >1} a_{ij}u_{1}^{i}u_{2}^{j} = 0.

as one verifies by checking in each of the charts Z(σ) ≃ A^{2}(C). Since we saw that every plane branch is similarly a deformation of the monomial curve with the same semi-group, which is defined by g binomial equations in variables u_{0}, …, u_{g}, adding to each binomial only monomials of higher weight, one is ready to believe that similarly, a regular fan in R_{≥0}^{g+1} which is compatible with the g hyperplanes corresponding to the g binomials will provide a toric map

π(Σ): Z(Σ) → A^{g+1}(C)

which is an embedded resolution not only for the monomial curve, but also for our original plane curve re-embedded in A^{g+1}(C) as was explained above. This is described in detail in [G-T] and generalized in [GP] to a much larger class of singularities.

This method of embedded resolution is quite different from the resolution by point blowing ups explained above, but it assumes that one knows the existence of a parametrization. The connection between the toric map and the sequence of point blowing ups is rather subtle (see [GP]); in the case g = 1 it is equivalent to the relation between finding approximations of a rational number by the reduced fractions of its continued fraction expansion and finding approximations by Farey series.

Complex curve singularities 55

So the deformation to the monomial curve also explains to us how to resolve the singularities, and it is perhaps the best description. Can we generalize it to higher dimensions?

10. Relation with topology

I refer to the lectures of Lê and to [B-K] for the Burau-Zariski topological interpretation of the characteristic sequence

(β_{0}, β_{1}, … β_{g})

as a characteristic of the iterated torus knot that one obtains upon intersecting the branch X with a sufficiently small sphere in C^{2} centered at the origin.

Given a germ of a reduced plane curve X, it has a decomposition X = ⋃_{i=1}^{r} X_{i} into branches; each branch has its characteristic sequence B(X_{i}), and as numerical characters of X, we have also the intersection numbers (X_{i}, X_{j})_{0} of distinct branches at 0.

If we remember that these intersection numbers are equal to the linking numbers in S^{3} of the knots corresponding to X_{i} and X_{j} and are therefore topological characters of the link X ∩ S_{ϵ}^{3}, since Milnor proved (see Lê’s lectures) that the curve X is homeomorphic to the cone with vertex 0 drawn on this link, we expect that the collection of the characteristic sequences of the branches and their intersection numbers may be a topological invariant of the curve X.

Let us define the local topological type of a germ of subspace of C^{N} as follows: Definition.- Two subspaces X_{1} and X_{2} of C^{N} are topologically equivalent at 0 if there exist neighbourhoods U and V of 0 in C^{N} and an homeomorphism ψ: U → V such that ψ(X_{1} ∩ U) = X_{2} ∩ V. Two germs at 0 of subspaces are topologically equivalent if they have representatives which are topologically equivalent at 0.

Theorem 10.1. (Zariski, Lejeune-Jalabert). Two germs of plane curves X = ∪_{i∈I}X_{i} and X′ = ∪_{i∈I′}X′_{i} are topologically equivalent if and only if there exists a bijection ϕ: I → I′ between their branches which preserves characteristics and intersection numbers, that is, satisfies

B(X′_{ϕ(i)}) = B(X_{i}) for i ∈ I, (X′_{ϕ(i)}, X′_{ϕ(j)})_{0} = (X_{i}, X_{j})_{0} for i ≠ j.

56 Bernard Teissier

Topological equivalence is less strict a relation than analytic (or even C^{1}) equivalence.

Let X_{1} and X_{2} each consist of four lines through the origin in C^{2}. According to the previous theorem, these two germs are topologically equivalent. However, if there was a germ et 0 of a C^{1} (and in particular analytic) isomorphism of C^{2} to itself, sending X_{1} to X_{2}, its tangent linear map at 0 would have to send X_{1} onto X_{2}. But two quadruplets of lines through 0 are linearly equivalent if and only if they have the same cross-ratio. If the slopes of the lines of X_{1} are a_{1}, b_{1}, c_{1}, d_{1}, and similarly for X_{2}, the cross ratios are

(\frac{a_{1} − a_{3}}{a_{1} − a_{4}} )(\frac{a_{2} − a_{4}}{a_{2} − a_{3}} )

and the numbers obtained by permutation. It is therefore easy to find examples where X_{1} and X_{2} are not C^{1}-equivalent.

In particular, in an analytic family of curves such as the surface in C^{3} with equation

(y − x)(y + x)(y − 2x)(y + tx) = 0

for small values of t, the fibers are all analytically inequivalent but topologically equivalent.

Theorem 10.2. Given two reduced germs of plane curves (X, 0) ⊂ (C^{2}, 0) and (X′, 0) ⊂ (C^{2}, 0) the following conditions are equivalent:

1) X and X′ are topologically equivalent,

2) There exists an integer d, a germ of curve (C, 0) ⊂ (C^{d}, 0) and two linear projections p, p′: C^{d} → C^{2}, both general for C at 0, and such that p(C) = X, p′(C) = X′,

3) There exists a one-parameter family of germs of plane curves that is a germ along {0}×U of a surface in C^{2} ×U, where U is a disk in C, say with equation f(x, y, u) = 0 and v, v′ ∈ U such that the germs of plane curve f(x, y, v) = 0, f(x, y, v′) = 0 are isomorphic to X, X′ respectively and all the germs f(x, y, t) = 0 have the same topological type for t ∈ U.

4) There exists a bijection from the set of branches of (X, 0) to the set of branches of (X′, 0) which preserves characteristic (Puiseux) exponents and intersection numbers.

5) The minimal embeded resolution processes of (X, 0) and (X′, 0) are ”the same” in the sense that one blows up at each step points with the same multiplicity.

Complex curve singularities 57

In fact, the theory of Lipschitz saturation, summarized in [T4], shows that, given the topological type of a germ of plane curve (X, 0), there exists a germ of a space curve (X^{s}, 0) ⊂ (C^{N}, 0), unique up to isomorphism, such that the germs of plane curves having the same topological type as (X, 0) are exactly, up to isomorphism, the images of (X^{s}, 0) by the linear projections (C^{N}, 0) → (C^{2}, 0) which are general for (X^{s}, 0).

11. Duality

A line in the projective space P^{2} is by definition a point in the dual projective space P̌^{2}.

Poncelet saw that given a nondegenerate conic Q, to any point P ∈ P^{2}, one can associate the polar curve of P with respect to Q, which is the line joining the points of contact with Q of the tangents to Q passing through P. We get in this way an isomorphism between P^{2} and its dual P̌^{2} We shall, however, refrain from identifying P^{2} and its dual in this way, as was done at that time.

The collection of points of P̌^{2} corresponding to the lines in P^{2} tangent to an algebraic curve C is an algebraic curve Č ⊂ P̌^{2}. Here we use the fact that one can say that a line is tangent to a curve C at a singular point x if its direction is a limit direction of tangents to C at non singular points tending to x.

A point x in P^{2} corresponds to a line x̌ in P̌^{2}; each point of this line represents a line in P^{2} which contains x, and the lines through x tangent to C correspond to the intersection points in P̌^{2} of the curve Č and the line x̌. So the class of the curve C, defined as the number of lines tangent to C at non singular points and passing through a given general point of P^{2}, is the degree m̌ of Č. Let us compute it:

Poncelet considered, following Monge, the polar curve (the terminology is his): Let

f(X, Y, X) = 0,

where f is a homogeneous polynomial of degree m, be an equation for C. The points of C where the tangent goes though the point of P^{2}(C) with coordinates (ξ, η, ζ) are on C and on the curve of degree m−1 with equation

P_{(ξ,η,ζ)}(f) = ξ \frac{∂f}{∂X} + η \frac{∂f}{∂Y} + ζ \frac{∂f}{∂Z} = 0

obtained by polarizing the polynomial f with respect to the point (ξ, η, ζ). If C is non singular, the points we seek are all the intersection points of C

58 Bernard Teissier

and P_{(ξ,η,ζ)}(C). By Bézout’s theorem, the number of these points counted with multiplicity is m(m−1), for every point (ξ, η, ζ) it is equal to m(m−1) if C has no singularities.

It is “geometrically obvious” that ˇ Č = C ; this is called biduality (it is completely wrong if we do geometry over a field of positive characteristic, so beware of what is “geometrically obvious”). If the curve Č had no singularities as well, the computation of degrees would give m(m − 1)(m^{2} − m − 1) = m, which holds only for m = 2. So if m > 2 the dual of a non singular curve has singularities ; for a general non singular curve, double points (a.k.a. nodes) of Č correspond to double tangents of C and cusps correspond to its inflexion points.

To understand biduality better, it becomes important to find the class of a projective plane curve with singularities, at least when these singularities are the simplest: nodes and cusps. This was done by Plücker and the formula for a curve with δ nodes and κ cusps is

m̌ = m(m − 1) − 2δ − 3κ.

One said that“ a node decreases the class by two, and a cusp by three” This is perhaps the first example of a search of numerical invariants of singularities.

One can compute the diminution of class provoked by an arbitrary plane curve singularity as follows: let f(x, y) = 0 be a local equation for the singular curve at a singular point x which we take as origin. The ideal j(f) = (\frac{∂f}{∂x}, \frac{∂f}{∂y})C\{x, y\} defines set-theoretically the origin, which is the only singular point (locally). Therefore it contains a power of the maximal ideal and it is a vector space of finite codimension μ(C, x) in C{x, y}. It is called the Milnor number of the singularity. Let m(C, x) be the multiplicity of C at 0, the order of the equation f(x, y). Then (see [T6]) the diminution of class due to the singularity is

Δ_{C,x} = μ(C, x) + m(C, x) − 1.

This means that for an arbitrary reduced projctive plane curve C of degree m we have the equality (generalized Plücker formula)

m̌ = m(m − 1) − Σ_{x∈C}Δ_{C,x}.

This number is also the local intersection number at x of the curve C and one of its “general local polar curves”, defined by α\frac{∂f}{∂x} + β \frac{∂f}{∂y} = 0 for geneal values of α, β. In this way, one can see that if we locally deform our singular curve to a non singular one, say by taking the equation f(x, y) = λ, the

Complex curve singularities 59

number of points where the tangent to this non singular curve has a fixed direction and which coalesce to 0 as λ → 0 is Δ_{C,x}; its is the number of possible tangents that are “absorbed” by the singular point. It is a remarkable fact that the “diminution of class” Δ_{C,x} depends only on the topological type of the germ of plane curve (C, x). In fact there is a formula to express it in terms of the Puiseux exponents of the branches of (C, x) and their local intersection numbers. The fact that the degree of the dual of a projective variety depends only on global characters like the degree and the “topology” of its singularities extends to an arbitrary singular projective variety if one makes the notion of topological type somewhat more stringent.

In the theory of algebraic curves, an important formula states that given an algebraic map f: C → C′ between non singular algebraic curves, which is of degree degf = d (meaning that for a general point c′ ∈ C′, f^{−1}(c′) consists of d points, and is ramified at the points x_{i} ∈ C, 1 ≤ i ≤ r, which means that near x_{i}, in suitable local coordinates on C and C′, the map f is of the form t ↦ t^{e_{i}+1} with e_{i} ∈ N, e_{i} ≥ 1. The integer e_{i} is theramification index of f at x_{i}. Then we have the Riemann-Hurwitz formula relating the genus of C and the genus of C′ via d and the ramification indices:

2g(C) − 2 = d(2g(C′) − 2) + ∑_{i} e_{i}

If we have a finite map f: C → C′ between possibly singular curves, it extends in a unique manner to a finite map of the same degree between their normalizations, to which we can apply the Riemann-Hurwitz formula to get a relation between the geometric genera of C and C′. If we apply this formula to the case where C is non singular and C′ = P^{1}, knowing that any compact algebraic curve is a finite ramified covering of P^{1}, we find that we can calculate the genus of C from any linear system of points made of the fibers of a map C → P^{1} if we know its degree and its singularities: we get

2g(C) = 2 − 2d + ∑ e_{i}

The ramification points x_{i} can be computed as the so-called jacobian divisor of the linear system, which consists of the singular points, properly counted, of the singular members of the linear system. In particular if C is a plane curve and the linear system is the system of its plane sections by lines through a general point x = (ξ : η : ζ) of P^{2}, the map f is the projection from C to P^{1} from x; its degree is the degree m of C and its ramification points are exactly the points where the line from x is tangent to C. Since x is general, these are simple tangency points, so the e_{i} are equal to 1, and

60 Bernard Teissier

their number is equal to the class m̌ of C; the formula gives

2g(C) − 2 = −2m + m̌ ,

thus giving for the genus an expression which is linear in the degree and the class, whereas our expression in terms of the degree alone is quadratic.

This is the first example of the relation between the “characteristic classes” (in this case only the genus) and the polar classes; in this case the curve itself, of degree m and the degree of the polar locus, or apparent contour from x, i.e. in this case the class m̌. The extension to a non singular projective algebraic variety in characteristic zero is due to Todd.

12. The polar curve

The (general) polar curve plays a much more important role in the study of the plane curve singularities than just giving the diminution of class by its intersection number with the curve at the singular point.

Given an equation f(x, y) = 0 for a germ reduced plane curve in (C^{2}, 0), let us denote by ℓ(x, y) a homogeneous linear form, i.e., the equation of a line through the origin. The we define a map

F_{ℓ}: (C^{2}, 0) → (C^{2}, 0)

by t_{0} = f(x, y), t_{1} = ℓ(x, y). The critical locus P_{ℓ} of this map is the local avatar of the polar curve which we saw in the previous section. More precisely if our germ of plane curve comes from a projective plane curve, then P_{ℓ} is the germ at 0 of the polar curve in the projective plane corresponding to the point at infinity in the direction of ℓ. Remark now that the image of P_{ℓ} by the map F_{ℓ} is a plane curve whose equations is given by the Fitting ideal of the algebra of P_{ℓ} as a C{t_{0}, t_{1}}- module, at least when the map (P_{ℓ}, 0) → (C^{2}, 0) induced by F_{ℓ} is finite, which is the case when ℓ is general. This image D_{ℓ} = F_{ℓ}(P_{ℓ}) is the discriminant of F_{ℓ}, and it lives in a plane with given coordinates t_{0}, t_{1}. The Newton polygon of D_{ℓ} in the coordinates t_{0}, t_{1} for a general choice of the linear form ℓ is independant of ℓ and is called the jacobian Newton polygon of f. M. Merle proved that if f(x, y) is irreducibule, the jacobian Newton polygon is a complete invariant of equisingularity of the curve f(x, y) = 0; it can be computed from the Puiseux exponents and determines them. The extension to the reducible case is due to E. Garcia Barroso (for all this, see [G]).

Complex curve singularities 61

The jacobian Newton polygon encodes the essence of the dynamics as λ goes to zero of the points of the non-singular curve f(x, y) = λ where the tangent is parallel to ℓ(x, y) = 0. Those are the points counting for the class of a non singular curve degenerating to our singular curve which are ”absorbed” by the singularity and so decrease the class.

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