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\begin{talk}[]{Bernard Teissier}
{Some aspects of the connection between toric geometry and resolution of singularities}
{Teissier, Bernard}

\noindent
We know from \cite{TE} that normal toric varieties over a field admit (non embedded) resolutions of singularities described by the regular refinements of their fan. The toric \emph{embedded} resolution of singularities for affine toric varieties over an algebraically closed field $k$ was proved in \cite{GP-T} and \cite{T-valuations}. The combinatorics works as follows: an affine toric variety $X_0\subset \A^N(k)$ over $k$ is defined by a prime binomial ideal $I_0=(u^{m^\ell}-\lambda_\ell u^{n^\ell})_{\ell\in L}$ in $k[u_1,\ldots ,u_N]$. The monomial $u^m$ corresponds to a point $m$ in the lattice $M\simeq \Z^N$, and $\lambda_\ell\in k^*$. The vectors $m^\ell-n^\ell\in M$ determine dual hyperplanes $H_\ell$ in the real vector space $N_\R$ generated by the dual lattice $N\simeq \check\Z^N$ of $M$. The intersections with the first quadrant of these hyperplanes determine a fan $\Sigma_0$ subdividing the fan whose maximal cone is the first quadrant. The strict transform of $X_0$ by the corresponding birational map $\pi(\Sigma_0)\colon Z(\Sigma_0)\to\A^N(k)$ of normal toric varieties is  the normalization of $X_0$. The strict transform of $X_0$ by a birational toric map $\pi(\Sigma)\colon Z(\Sigma)\to\A^N(k)$ corresponding to a regular fan $\Sigma$ subdividing $\Sigma_0$ is non singular and transversal to the toric boundary. Such subdivisions provide embedded pseudo\footnote{This means that the restriction over the non singular part is not necessarily an isomorphism.} resolutions of $X_0$. The fan $\Sigma$ can be chosen so as to contains the regular faces of the \emph{weight cone} $\beta=\R_{\geq 0}^N\cap(\bigcap_\ell H_\ell)$, and then $\pi(\Sigma)$ is an embedded resolution.\par
One may wonder whether such toric maps also (pseudo) resolve the spaces obtained by suitable deformations of the binomial equations. This question comes from the basic observation of \cite{T-valuations}: Given a local integral domain $R$ with maximal ideal $m$ and a rational valuation of $R$ corresponding to an inclusion $R\subset R_\nu$ of $R$ in a valuation ring $R_\nu$ of its field of fractions, such that $m_\nu\cap R=m$ and $R/m\to R_\nu/m_\nu$ is an isomorphism, we have a faithfully flat specialization of ${\rm Spec}R$ to the affine toric variety (which may be of infinite embedding dimension) corresponding to the associated graded ring ${\rm gr}_\nu R=\bigoplus_{\phi\in\Phi} \Pp_\phi/\Pp^+_\phi$ of $R$ with respect to the filtration associated to $\nu$, where $\Pp_\phi=\{x\in R\vert\nu(x)\geq \phi\},\ \Pp^+_\phi=\{x\in R\vert\nu(x)>\phi\}$. The fact that $\nu$ is a rational valuation implies that ${\rm gr}_\nu R$ is a $k$-algebra and each homogeneous component is a vector space of dimension $1$ over $k$. There is therefore a presentation ${\rm gr}_\nu R=k[(U_i)_{i\in I}]/(U^{m^\ell}-\lambda_\ell U^{n^\ell})_{\ell\in L}$ where $U^m$ denotes a monomial, $\lambda_\ell\in k^*$, the sets $I$ and $L$ may be infinite, but countable.\par
We note that the degrees which actually appear in the graded algebra are the valuations of the elements of $R$, which form a subsemigroup of the semigroup $\Phi_+\cup\{0\}=(R_\nu\setminus\{0\})^{mult.}/\{units\}$ of non negative elements of the (totally ordered) value group $\Phi$ of $\nu$. In fact ${\rm gr}_\nu R$ is isomorphic to the semigroup algebra over $k$ of the semigroup $\Gamma=\nu(R\setminus \{0\})$. If $R$ is noetherian the semigroup $\Gamma$ is well ordered and therefore has a unique minimal system of generators, indexed by an ordinal, which is at most $\omega^h$ where $h$ is the (archimedian, or real) rank of the value group. By transfinite induction one defines $\gamma_{i+1}$ as the smallest non zero element of $\Gamma$ which is not in the semigroup generated by the previous ones.\par
Let us concentrate on the case where the semigroup $\Gamma$ is finitely generated and $R$ is a local equicharacteristic and complete noetherian domain with an algebraically closed residue field $k$. Pick and fix a field of representatives $k\subset R$. Then $R$ appears as an \emph{overweight} deformation of its associated graded ring, in the sense of \cite{OWR2}: there is a continuous and surjective map of $k$-algebras
$$k[[u_1,\ldots, u_N]]\stackrel \pi\longrightarrow R,\ {\rm determined\  by} \ \ u_i\mapsto \xi_i,$$
for any choice of elements $\xi_i\in R$ whose valuations are the minimal generators of the semigroup $\Gamma$ or equivalently are such that their initial forms minimally generate the $k$-algebra ${\rm gr}_\nu R$. Giving to $u_i$ the weight $\gamma_i=\nu(\xi_i)\in\Gamma\subset\Phi_+\cup\{0\}$ determines a weight $w$ on $k[[u_1,\ldots, u_N]]$, with its filtration by weight and the associated graded ring ${\rm gr}_w k[[u_1,\ldots, u_N]]\simeq k[U_1,\ldots, U_N]$, now graded by the weight: ${\rm deg}U_i=\gamma_i$. Moreover the valuation ideals of $R$ are the images by $\pi$ of the weight ideals of $k[[u_1,\ldots, u_N]]$ and so the map $\pi$ induces a surjection of graded $k$-algebras
$$ k[U_1,\ldots ,U_N] \stackrel {{\rm gr}_w\pi}\longrightarrow  {\rm gr}_\nu R, \ {\rm determined\  by} \ \ U_i\mapsto {\rm in}_\nu\xi_i,$$ whose kernel is a binomial ideal $(u^{m^\ell}-\lambda_\ell u^{n^\ell})_{\ell\in L}$; it is essentially the presentation of the semigroup algebra of $\Gamma$ over $k$ which corresponds to an affine toric variety $X_0$.  By flatness the kernel of $\pi$ is generated by series $F_\ell=u^{m^\ell}-\lambda_\ell u^{n^\ell}+\sum_pc^{(\ell)}_pu^p$ with $c^{(\ell)}_p\in k,\ w(u^p)>w(u^{m^\ell})=w(u^{n^\ell}) $, for $\ell\in L$, a finite set. Let us call $X$ the formal subspace of $\A^N(k)$ defined by the ideal $I=(F_\ell)_{\ell\in L}$; it is an \emph{overweight deformation} of the affine toric variety $X_0$.\par
For a regular fan $\Sigma$ with support the first quadrant of $\check\R^N$, the corresponding birational toric map $Z(\Sigma)\rightarrow \A^N(k)$ is described in each chart $Z(\sigma)$ corresponding to a maximal cone $\sigma=\langle a^1,\ldots ,a^N\rangle$ of $\Sigma$, where $a^j\in N$,  by
$$\begin{array}{lr}
\ \ \ \ \ \ \ u_1=y_1^{a^1_1}\ldots y_N^{a^N_1}\\
\ \ \ \ \ \ \ .\ \ \ \ \ \ \ \ .\ \ \ \ \\
\ \ \ \ \ \ \ .\ \ \ \ \ \ \ \ .\ \ \ \ \\
\ \ \ \ \ \ \ .\ \ \ \ \ \ \ \ .\ \ \ \ \\
\ \ \ \ \ \ \ u_N=y_1^{a^1_N}\ldots y_N^{a^N_N}
\end{array}$$
and the valuation $\nu$ of $R$ picks a point in the strict transform of $X$. A combinatorial argument explained in \cite{overweight} shows that one can find regular fans $\Sigma$ subdividing the fan $\Sigma_0$ corresponding to the initial binomials of the $F_\ell$, and such that for appropriate $\sigma\in \Sigma$ the transforms of the $F_\ell$ can be written 
$$\small{\begin{array}{lr} F_\ell\circ\pi(\sigma)=\\ y_1^{\langle a^1,n^\ell\rangle}\ldots y_N^{\langle a^N,n^\ell\rangle}(y_1^{\langle a^1,m^\ell-n^\ell\rangle}\ldots y_N^{\langle a^N,m^\ell-n^\ell\rangle}-\lambda_\ell+\sum_pc^{(\ell)}_py_1^{\langle a^1,p-n^\ell\rangle}\ldots y_N^{\langle a^N,p-n^\ell\rangle}).\end{array}}$$ 
The point is to find fans for which the inequalities $w(u^p)>w(u^{n^\ell})$ induce inequalities $\langle a^i,p-n^\ell\rangle>0$. The largest torus-invariant charts of $Z(\Sigma)$ in which the strict transform meets the toric boundary correspond to cones $\sigma$ of $\Sigma$ whose intersection with the weight cone $\beta$ is of maximal dimension $r={\rm dim}R$. The variables $y_{i_j},\ 1\leq j\leq r$ corresponding to the vectors $a^{j_i}\in\beta$ do not appear in the transformed binomials $y_1^{\langle a^1,m^\ell-n^\ell\rangle}\ldots y_N^{\langle a^N,m^\ell-n^\ell\rangle}-\lambda_\ell$ and can be taken as local coordinates on the strict transform of $X$. In fact, \emph{at the point picked by the valuation}, this strict transform is a deformation of the strict transform of $X_0$ and hence non singular. In summary:\par\noindent
Theorem:\emph{ Given a rational valuation $\nu$ on a complete equicharacteristic local domain $R$ with an algebraically closed residue field $k$, if the semigroup of values\break $\nu(R\setminus\{0\})$ is finitely generated, say by $N$ generators, there is a continuous surjection $k[[u_1,\ldots, u_N]]\stackrel \pi\longrightarrow R$ such that some of the toric modifications of $\A^N(k)$ in the coordinates $u_i$ which resolve the singularities of the toric variety corresponding to ${\rm gr}_\nu R$ also produce an embedded local uniformization of the valuation $\nu$ on the space $X\subset \A^N(k)$ corresponding to $R$.}\par\medskip
In the situation of the theorem, by flatness of the deformation, the valuation $\nu$ is Abhyankar, which means in this case that the Abhyankar inequality ${\rm dim}{\rm gr}_\nu R\leq {\rm dim}R$ (see \cite{T-valuations}) is an equality. Since local uniformization for Abhyankar valuations of algebraic function fields has been proved by Knaf and Kuhlmann in \cite{KK1}, it is natural to ask whether in general the Abhyankar condition implies that the semigroup $\Gamma$ is finitely generated. An attempt to prove this is in progress. Combined with the theorem above it would have as consequence that the Abhyankar valuations are exactly the quasi-monomial ones, a fact proved by Cutkosky for valuations of rank one using embedded resolution of singularities (see \cite{ELS}, Prop. 2.8).




\begin{thebibliography}{99}

\bibitem{ELS} L.~Ein, R. Lazarsfeld, and K. Smith, \emph{Uniform approximation of valuation ideals in smooth function fields}, Amer. J. Math. 125 (2003), no. 2, 409-440. 

\bibitem{TE}
G.~Kempf, F.~F. Knudsen, D.~Mumford, and B.~Saint-Donat, \emph{Toroidal
  embeddings. {I}}, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag,
  Berlin, 1973.

\bibitem{GP-T}
P.~D. Gonz\'alez~P\'erez and B.~Teissier, \emph{Embedded resolutions of not
  necessarily normal affine toric varieties}, C.R. Math. Acad. Sci. Paris
  \textbf{334} (2002), no.~5, 379--382.
  
 \bibitem{KK1} Hagen Knaf,  Franz-Viktor Kuhlmann, \emph{ Abhyankar places admit local uniformization in any characteristic},  Ann. Sci. \'Ecole Norm. Sup. (4)  38  (2005),  no. 6, 833--846.

 \bibitem{T-valuations}
B.~Teissier, \emph{Valuations, deformations, and toric geometry}, Valuation theory
  and its applications, {V}ol. {II} ({S}askatoon, {SK}, 1999), Fields Inst.
  Commun., vol.~33, Amer. Math. Soc., Providence, RI, 2003, 361--459.
  
\bibitem{OWR2}
B. Teissier, \emph{Overweight deformations of weighted affine toric varieties}, Oberwolfach Workshop on Toric geometry, January 2009. Oberwolfach Reports, Vol. 6, No.1, 2009. European Math. Soc. Publications. 
  
\bibitem{OWR1}
 B. Teissier, \emph{A viewpoint on local resolution of singularities}. Oberwolfach Workshop on Singularities, September 2009. Oberwolfach Reports, Vol. 6, No. 3, 2009. European Math. Soc. Publications.

   \bibitem{overweight}
B.~Teissier, \emph{Overweight deformations of affine toric varieties and local uniformization}, submitted.


 \end{thebibliography}

\end{talk}


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