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\begin{document}
\title
{A viewpoint on resolution of singularities\\
Liverpool Bruce60Wall75, June 2012
}

\author{Bernard Teissier} 
\maketitle
%\begin{document}
\section{The present}
All approaches to resolution of singularities follow the ideas of Hironaka's fundamental paper, as modified by Villamayor to put the idealistic exponents, or basic objects, at the center of the process, as the only objects which need to be resolved (see \cite{Cut}, \cite{K}).
The basic attitude is to compare a singular space with a non singular one or an ideal with the ideal generated by independent linear forms. The basic invariants one tries to decrease by blowing-ups are the multiplicity (which measures the difference between a singular point of a space and a point on a non singular space of the same dimension, or the \emph{order} at a point of an ideal on a non singular space, which measures the obstruction to finding in the ideal the equation of a hypersurface which is non singular at the point considered.  \par
The main problem in positive characteristic is the non-existence of "hypersurfaces of maximal contact" which reflects in part the unruly behavior of multiplicity and order in positive characteristic.\par
In the last years I have been led to try to prove local uniformization (a very local version of resolution) by a completely different method, in which the basic idea is to compare a given singular germ \emph{by deformation} with a space whose resolution is easy and blind to the characteristic. The spaces in question are affine toric varieties, which are defined by prime binomial ideals. I refer to \cite{GP-T1} for their toric \emph{embedded} resolution and to \cite{B-M} for the proof of a canonical embedded resolution by composition of blowing-ups with equivariant non singular centers.\par\noindent
The divisors of the toric boundary (=complement of the dense torus) of the affine toric variety collectively play the role of a maximal contact divisor, in a combinatorially complicated way.
\par
Let $R$ be a noetherian excellent equicharacteristic local domain with an algebraically closed residue field $k=R/m$. Let $\nu$ be a valuation on $R$, corresponding to an inclusion $R\subset R_\nu$ of $R$ in a valuation ring of its field of fractions. We may assume that $R_\nu$ dominates $R$ in the sense that $m_\nu\cap R=m$ and that the residual injection $R/m\hookrightarrow R_\nu/m_\nu$ is an isomorphism. This corresponds to the fact that the point picked by $\nu$ in all schemes birationally dominating $\hbox{\rm Spec}R$ by a proper map is a \emph{closed} point.\par
To this situation is naturally associated an affine toric variety, of finite Krull dimension but possibly of infinite embedding dimension. It corresponds to the \emph{associated graded ring} of the ring $R$ with respect to the filtration defined by the valuation. This filtration is indexed by a well ordered semigroup, the semigroup of values which the valuation takes on the ring $R$. Its ordinal type is $\leq \omega^h$ where $h$ is the (real, or archimedian) rank of the valuation. This graded ring turns out to admit a presentation as a quotient of a polynomial ring over $k$, whose variables are in bijection with the well ordered set consisting of a minimal system of generators of the value semigroup $\Gamma=\nu(R\setminus\{0\})$. The ideal defining the quotient is generated by a possibly infinite set of binomials in the variables described above. So the associated graded ring is essentially the semigroup algebra of $\Gamma$.\par
In many important cases (see \cite{Te1}, \cite{Te3}), one can check that there exists a formal embedding of $(\hbox{\rm Spec}R, m)$ in an affine space $({\mathbf A}^N(k),0)$ with the following properties: there is a system of coordinates such that the intersection of $(\hbox{\rm Spec}R, m)$ with the torus $T^N(k)$ consisting of the complement of the coordinate hyperplanes is dense, and a birational map of toric varieties $Z\to {\mathbf A}^N(k)$ with $Z$ regular, which is equivariant with respect to $T^N(k)$ and such that the strict transform of $(\hbox{\rm Spec}R, m)$ in $Z$ is non singular and transversal to the toric boundary of $Z$ \emph{at the point of this strict transform picked by the valuation $\nu$}.\par\noindent
Such a result is a constructive form of local uniformization, at least if one can effectively construct the embedding.\par
In the case of plane branches (see \cite{G-T}) and more generally of quasi-ordinary hypersurfaces (see \cite{GP}), the smallest embedding with this property can be explicitly constructed in characteristic zero from (generalized) Puiseux expansions. \par
Since it seems much easier to glue up the embeddings corresponding to various valuations (by compactness of the Zariski-Riemann manifold a finite number suffices) than to glue up \emph{\`a la} Zariski various birational models, this led me to ask in \cite{Te2} the following:\par\noindent
\textbf{Question:} \emph{Given a noetherian excellent equicharacteristic local domain $R$ with an algebraically closed residue field $k=R/m$, does there exist a formal embedding of $(\hbox{\rm Spec}R, m)$ with a toric birational map of toric varieties $Z\to {\mathbf A}^N(k)$ in appropriate coordinates on ${\mathbf A}^N(k)$ such that the strict transform of $\hbox{\rm Spec}R$ in $Z$ is non singular and transversal to the toric boundary at each point mapped to the closed point $m$ of $\hbox{\rm Spec}R$?}\par\noindent One may ask that in addition the singular locus of $\hbox{\rm Spec}R$ should be the union of intersections with $\hbox{\rm Spec}R$ of sets of coordinate hyperplanes in this new embedding. The map from the strict transform to $\hbox{\rm Spec}R$ is then an isomorphism outside of the singular locus. I think of this as a generalization of the condition of non-degeneracy with respect to a Newton polyhedron.\par
This would imply local resolution of singularities and the difficulty is moved from the study of the behaviour of the order of ideals under certain blowing ups to the search of functions in $R$ having very special properties. The simplest non trivial example is the plane curve $(y^2-x^3)^2-x^5y=0$ which can be resolved by a single toric modification of the ambient space only after being embedded in ${\mathbf A}^3(k)$ by the functions $x,y,y^2-x^3$. See \cite{G-T}, \cite{Te1} and \cite{Te3}.\par\medskip
After hearing me mention this in January 2009 at the Oberwolfach Workshop on Toric Geometry (see \cite{Te3}), Jenia Tevelev kindly sent me a proof of the following:
\begin{theorem} {\rm (Tevelev)} Let $k$ be an algebraically closed field of characteristic zero and let $X\subset {\mathbf P}^n(k)$ be a projective algebraic variety. Then, for a sufficiently high order Veronese reembedding $X\subset {\mathbf P}^N(k)$ one can choose projective coordinates $z_0:\ldots :z_N$ such that, if $T^N(k)$ is the torus $(k^*)^N$ consisting of the complement of the coordinate hyperplanes in ${\mathbf P}^N(k)$, then:
\begin{itemize} 
\item The intersection of $X$ with $T^N(k)$ is dense in $X$,
\item There exists a nonsingular toric variety $Z$ and an equivariant map $Z\to {\mathbf P}^N(k)$ such that the strict transform of $X$ is non singular and transversal to the non dense toric orbits in $Z$.
\item The map $Z\to {\mathbf P}^N(k)$ above may be taken to be the composition of equivariant blowing-ups with non singular centers.\end{itemize}
\end{theorem}
The proof uses resolution of singularities and answers the question in characteristic zero for algebraizable singularities while of course one would hope to prove local resolution in the manner I have described. Still, it is very encouraging.\par
\section{The future}
This is part of a general trend in singularity theory which I find fascinating. It has several facets.
\begin{itemize} 
\item One is the study of toric degenerations, of which what I have just described is an example. There are now many examples in global situations where algebraic varieties degenerate to toric varieties. Here it is important, locally as well as globally, to consider all toric varieties and not just the normal ones. See \cite{GP-T1} and \cite{GP-T2}. In particular an affine toric variety over a field $k$ is the spectrum of the semigroup algebra of a commutative cancellative torsion free semigroup, usually supposed to be of finite type. \par\noindent
\item Another is the study of infinite dimensional spaces such as valuation spaces, Berkovich spaces, arc spaces. There are spectacular results in the study of plurisubharmonic functions (see \cite{B-F-J1}) from the valuative point of view, and also substantial progress in intersection theory from that viewpoint (see \cite{B-F-J2}). The new challenge is to do analysis on valuation spaces, as Favre-Jonsson do, defining  Monge-Amp\`ere operators on valuation spaces and showing that their properties correspond to deep facts of algebraic geometry. The geometries of valuation spaces and Berkovich spaces begin to be understood, even when they are associated to singular spaces (see \cite{D} for a recent account) , but we still have to study, for example, their behavior under deformations that are equisingular in some sense, or the behavior of their fiber over points related by a toric map of the ambient space. The same questions are pertinent for arc and jet spaces at a point. \par\noindent
\item The geometry of valuation spaces and Berkovich spaces is illuminated by the connexion with the tropical geometry initiated by Oleg Viro and Grisha Mikhalkin. The process of tropicalization applies to a closed subvariety of the torus in a toric variety, in the beginning affine or projective space. Sam Payne showed in \cite{P} the importance of considering all embeddings of a given variety in toric varieties which are such that the intersection with the torus is dense. Conversely, Tevelev uses in an essential way in \cite{Tev} the concept of \emph{tropical compactification}. All this contains generalizations of the ideas of Newton polyhedron (over a valued field) and of non degeneracy. \par\noindent Now affine toric varieties are the simplest objects from the viewpoint of tropicalization and their geometry is combinatorially described by the semigroup to which they correspond, 
\end{itemize}
I believe that in the next few years the interactions between these topics will continue to flourish and provide very powerful tools, not only for resolution of singularities but for the understanding of singularities. This means in particular the relations between the  resolution simplicial complex (generalizing the dual graph of surface singularities) encoding part of the combinatorics of the resolutions of a singularity and appropriate systems of equations (for appropriate embeddings) of the singularity. It also means the relations between these equations and the topology of the singularity. The papers quoted in this section, together with \cite{P-S}, support this belief.

\begin{thebibliography}{99}

\bibitem{B-M} Bierstone, Edward; Milman, Pierre D. \emph{Desingularization of toric and binomial varieties},  J. Algebraic Geom.  15  (2006),  no. 3, 443--486.

\bibitem{B-F-J1} Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias, \emph{Valuations and plurisubharmonic singularities},  Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 449-494.

\bibitem{B-F-J2} Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias, \emph{Differentiability of volumes of divisors and a problem of Teissier}, J. Algebraic Geom. 18 (2009), no. 2, 279-308

\bibitem{C-P 1} V. Cossart and O. Piltant, \emph{Resolution of singularities of threefolds in positive characteristic, I ,Reduction to local uniformization on Artin-Schreier and purely inseparable coverings},  J. Algebra  320  (2008),  no. 3, 1051-1082.

\bibitem{C-P 2} V. Cossart and O. Piltant,  \emph{Resolution of singularities of threefolds in positive characteristic. II},  J. Algebra  321  (2009),  no. 7, 1836-1976.

\bibitem{Cut}
S. D. Cutkosky, \emph{Resolution of singularities}, Graduate Studies in Math. vol. 63, AMS 2004.

\bibitem{D} A.~Ducros, \emph{Espaces de Berkovich, polytopes, squelettes. et th\'eorie des mod\`eles}. arXiv:1203.6498v3 [math.AG] 12 Jul 2012.

\bibitem{F-J} Favre, Charles; Jonsson, Mattias, \emph{Valuative analysis of planar plurisubharmonic functions}, Invent. Math. 162 (2005), no. 2, 271-311.

\bibitem{G-T}  R. Goldin and B. Teissier, \emph{Resolving singularities of plane analytic branches with one toric morphism}, Resolution of singularities (Obergurgl, 1997),  315--340, Progr. Math., 181, Birkhäuser, Basel, 2000.

\bibitem{GP}
P.~Gonz\'alez P\'erez, \emph{Toric embedded resolutions of quasi-ordinary hypersurfaces},
Annales Inst. Fourier (Grenoble) 53, (2003), 1819-1881.

\bibitem{GP-T1}
P.~Gonz\'alez P\'erez et B. Teissier \emph{Embedded resolutions of non necessarily normal affine toric varieties},
Comptes-tendus Acad. Sci. Paris, Ser.1, 334, (2002), 379-382.

\bibitem{GP-T2} P.~Gonz\'alez P\'erez et B. Teissier \emph{Toric Geometry and the Semple-Nash modification} To appear in Revista de la Real Academia de Ciencias Exactas, F\'isicas y Naturales (Serie A - Matem\'aticas), Proceedings of the Hironaka80 Conference, Tordesillas 2011.

\bibitem{K} J. Koll\'ar, \emph{ Lectures on resolution of singularities}, Annals of Mathematics Studies, 166. Princeton University Press, Princeton, NJ, 2007.

\bibitem{P} S. Payne, \emph{Analytification is the limit of all tropicalizations}, Math. Res. Lett. 16 (2009), no. 3, 543-556.

\bibitem{P-S} P. Popescu Pampu and D.A. Stepanov, \emph{local tropicalization}, arXiv:1204.6154v1.

\bibitem{Te1}
B.~Teissier, \emph{Valuations, deformations, and toric geometry},
Valuation Theory and its applications, Vol. II, Fields Inst. Commun. 33, AMS., Providence, RI., (2003), 361-459.

\bibitem{Te2}
B.~Teissier, \emph{Monomial ideals, binomial ideals, polynomial ideals},
Trends in Commutative Algebra, MSRI publications, Cambridge University Press (2004), 211-246.

\bibitem{Te3}  B.~Teissier, \emph{Overweight deformations of weighted affine toric varieties}, Oberwolfach workshop on Toric Geometry, January 2009, Oberwolfach Report OWR 2009 01.

\bibitem{Tev} 
J. Tevelev, \emph{On a question of Bernard Teissier}. letter to the author, January 2009.

\end{thebibliography}






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 (CNRS, Institut math\'ematique de Jussieu, UMR 7586 du CNRS\\ 175 Rue du Chevaleret, 75013 Paris\\teissier@math.jussieu.fr )
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