Valuations, deformations, and toric geometry. (English) Zbl 1061.14016

Kuhlmann, Franz-Viktor (ed.) et al., Valuation theory and its applications. Volume II. Proceedings of the international conference and workshop, University of Saskatchewan, Saskatoon, Canada, July 28–August 11, 1999. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3206-9/hbk). Fields Inst. Commun. 33, 361-459 (2003).
This paper is a program (the author points out two difficult points left without proofs for the moment) to solve the problem of local uniformization in characteristic \(p>0\). The way chosen by Teissier is completely new and goes through the unknown territory of non Noetherian \(k\)-algebras. So the author had to prove plenty of new and technical results. The reviewer is optimistic about the successful completion of the program.
Definition. Let \(X\) be a complete variety over a field \(k\) and let \(\nu\) be a valuation of a field \(L\), which is an extension of the function field \(K\) of \(X\). Let \(R_\nu\subset L\) be the valuation ring of \(\nu\), and denote by \(m_\nu\) its maximal ideal. A point \(x\in X\) is a center of \(\nu\) if \({\mathcal O}_{X,x}\subset R_{\nu}\) and \(m_{X,x}={\mathcal O}_{X,x}\cap m_{\nu}\). We say then that \(R_{\nu}\) dominates \({\mathcal O}_{X,x}\).
The valuative criterion of properness says that if \(\nu\) is 0 on \(k^*\) (\(\nu\) is then said to be a \(L/k\)-valuation), then \(\nu\) has a unique center in \(X\).
The uniformization problem: Given \(X\) a \(k\)-projective variety, \(\nu\) a \(K/k\)-valuation, find a projective model (a projective variety birationnally equivalent to \(X\)) of \(X\) where the center of \(\nu\) is regular.
This problem was solved by O. Zariski [Ann. Math. (2) 40, 639–689 (1939; Zbl 0021.25303)] in the characteristic 0 case. The characteristic \(p\) case still resists. By the quasi-compactness of the Riemann-Zariski surface of \(X\), if this problem is solved, there will exist a finite numbers of projective varieties \(Y_1,\dots,Y_n\) birational to \(X\), with \(Y_i \to X\) projective such that for every \(L/k\)-valuation \(\nu\), there is some \(i\), \(1\leq i \leq n\) such that the center of \(\nu\) in \(Y_i\) is regular.
So we will be rather close to a desingularization of \(X\): this is Zariski’s strategy to prove desingularization of algebraic varieties.
Now to the paper at hand:
Main Idea: specialize the singularity \(R={\mathcal O}_{X,x}\subset R_\nu\) to a toric singularity (not necessarily normal), which turns out to be of finite Krull dimension but possibly of infinite embedding dimension, and show that one can partially resolve this toric singularity in an embedded manner in such a way that this resolution will extend to an embedded local uniformization of \(\nu\) on \(R\). This means a birational transformation \(R\subset R'\) induced by a regular monomial transformation on a system of generators of the maximal ideal of \(R\), such that \(R'\) is a regular local ring also dominated by \(R_\nu\).
Since the resolution of toric varieties by toric maps is a combinatorial fact which is blind to the characteristic, this will give a characteristic-free proof of local uniformization (over an algebraically closed field).
In fact two of the results proved in the paper are that in finite dimensions toric resolutions of the non-normal toric varieties exist and that if \(R\) is complete, and the graded \(k\)-algebra \(\text{ gr}_{\nu}R\) (see below) is finitely generated, the extension of such toric resolutions to uniformization of the valuation also exists. The major difficulties arise when trying to reduce the general case to this case (the simple case).
The author is led to use a new family of modifications: toric blowings-up which are related to a specific torus-action and which are not permissible. R. Goldin and B. Teissier [in: Resolution of singularities. Progr. Math. 181, 315–340 (2000; Zbl 0995.14002)] completed this program for plane analytically irreducible curves by re-embedding them in a higher-dimensional space in such a way that they then degenerate to the monomial curve with the same semigroup, which is a toric variety.
The hypothesis \(k\) algebraically closed allows the author to look only at rational valuations, i.e. we may suppose that \(k \simeq {R_{\nu} \over {m}_{\nu}}\) (part. 3).
Notations and Definitions: The graded ring associated to \((R,\nu)\) is \[ \text{gr}_{\nu}R=\bigoplus _{\phi \in \Phi} {\mathcal P}_{\phi}(R)/{\mathcal P}_{\phi}^+(R) \] where \(\Phi\) is the value group of \(\nu\), \({\mathcal P}_{\phi}(R)=\{x\in R|\nu(x)\geq \phi\}\), \({\mathcal P}_{\phi}^+(R)=\{x\in R\mid \nu(x)> \phi\}\). The valuation algebra associated to \((R,\nu)\) is \[ {\mathcal A}_{\nu}(R)=\bigoplus _{\phi \in \Phi} {\mathcal P}_{\phi}(R)v^{-\phi} \subset R[v^\Phi]. \] Here \(\Phi\) is the group of values of the valuation \(\nu\) and \(R[v^\Phi]\) is the group algebra of \(\Phi\) with coefficients in \(R\).
Propositions 2.2 and 2.3 (abridged by the reviewer): Assume that \(\nu\) is a rational valuation of \(R\) and that \(R\) contains a field of representatives of \(k=R/m\).
a) The natural map \( {\mathcal A}_{\nu}(R) \to \text{gr}_{\nu}R\) defined by \(x_{\phi}v^{-\phi} \to x_{\phi}\bmod {\mathcal P}_{\phi}^+(R)\) induces an isomorphism \( {\mathcal A}_{\nu}(R)/((v^\phi)_{\phi\in \Phi_+}) {\mathcal A}_{\nu}(R)) \to \text{gr}_{\nu}R\).
b) The graded \(k\)-algebra \(\text{gr}_{\nu}R\) is a quotient of a polynomial ring in countably many variables over \(R\) by an ideal generated by binomials: \[ \text{gr}_{\nu}R\simeq k[(U_i)_{i\in I}]/(U^m-\lambda_{m,n}U^n)_{(m,n)\in E} , \] where \(I\) and \(E\) are countable sets, \(U^m=U_{i_1}^{m_{i_1}}\dots U_{i_k}^{m_{i_k}}\) and \(\lambda_{m,n}\in k^*\).
c) The natural inclusion \(R[v^{{\Phi}_+}] \to {\mathcal A}_{\nu}(R)\) obtained by considering the part of negative degree induces an isomorphism \((v^{{\Phi}_+})^{-1})R[v^{{\Phi}_+}] \simeq (v^{{\Phi}_+})^{-1}) {\mathcal A}_{\nu}(R)\), furthermore the composed map \(k[v^{{\Phi}_+}] \to R[v^{{\Phi}_+}]\to {\mathcal A}_{\nu}(R)\) is faithfully flat.
d) Given an homomorphism \(c: \Phi \to k^*\), there is a natural surjection \( {\mathcal A}_{\nu}(R)\to R\) whose kernel is generated by the \((v^\phi-c(\phi))_{\phi\in \Phi}\).
The ingredients are there: the basis of the deformation is \(\text{Spec}~k[v^{{\Phi}_+}]\), the family is \(\text{Spec}~{\mathcal A}_{\nu}(R)\), the special fiber is \( \text{Spec}~\text{gr}_{\nu}R\), which is binomial, the “general fiber” of the faithfully flat family is isomorphic to \(\text{Spec} R\) over \((v^{{\Phi}_+})^{-1})k[v^{{\Phi}_+}]\) by c), and d) produces an explicit fiber isomorphic to \(R\) by the surjection \( {\mathcal A}_{\nu}(R)\to R\). One can take take \(c(\phi)=1\) \(\forall\;\phi\in \Phi\).
The fact that \(\text{gr}_{\nu}R\) is not finitely generated in general, even if \(R=k[[x,y]]\), is a basic difficulty of valuation theory.
Non Abhyankar valuations: Let us point out a tremendous pathology: in his thesis, Piltant proved that the dimension of \(\text{gr}_{\nu}R\) is equal to the rational rank of the valuation (\(=\dim_{\mathbb Q} \Phi \otimes \mathbb Q\)) and, if Abhyankar’s inequality is strict (non Abhyankar valuations), \(\dim(\text{gr}_{\nu}R)<\dim(R)\). This means that we have a faithfully flat family where the special fiber has a dimension strictly smaller than the generic fiber. Nothing is wrong: we are in the unknown territory of non Noetherian algebras, see p. 380.
In some sense, this pathology soothes the specialists: from their experiences, they note that the valuations where Abhyankar’s inequality is strict are extremely difficult to manage. This dimension pathology is the first objective argument to explain that very specific difficulty. The abyssal phenomenon below illustrates how the combinatorics of a toric variety of dimension \(< \dim R\) can describe local uniformization on \(R\) when it is complete.
Scalewise completion: Let’s go into the unknown territory. To show that a suitable partial resolution of the special fiber \(\text{gr}_{\nu}R\) extends to a local uniformization of the valuation \(\nu\) on the general fiber \(R\), the author needs equations of the family \( {\mathcal A}_{\nu}(R)\) in some regular space. One first needs suitable coordinates, and for that one needs the ring to be complete in some sense (think of Cohen’s theorem). However completion with respect to a valuation is much more subtle than with respect to an adic topology. The author conjectures the existence of a scalewise completion \(\widehat R^{(\nu)}\) of \(R\) which is a quotient \( \widehat R^{(\nu)}=\widehat R^m /H\) of the \(m\)-adic completion of the local ring \(R\), where \(H\subset \widehat R^m\) is a prime ideal such that \(H\cap R=(0)\). This quotient must have the property that the valuation \(\nu\) extends to a valuation \(\widehat\nu\) of \(\widehat R^{(\nu)}\) in such a way that the inclusion \(R\subset \widehat R^{(\nu)}\) induces a birational morphism \(\text{gr}_{\nu}R \to \text{gr}_{\widehat\nu}\widehat R^{(\nu)}\).
Teissier conjectures that this is possible provided that \(R\) is excellent, which is the case if \(R\) comes from an algebraic variety over \(k\). It is proved when \(\nu\) is of rank one.
If one assumes this to be true, the author proves that given a field or representatives \(k\subset \widehat R^{(\nu)}\) and a choice of elements \(\eta_j\in \widehat R^{(\nu)}\) whose images \(\overline \eta_j\in \text{gr}_{\widehat\nu}\widehat R^{(\nu)}\) form a minimal system of generators of that \(k\)-algebra, there is a surjection \[ \widehat{k[(w_j)_{j\in J}]} \to\widehat R^{(\nu)},\qquad w_j\mapsto \eta_j, \] where the left-hand side is a scalewise completion of the polynomial ring (in general it is not Noetherian); this surjection is such that if one takes in \(\widehat{k[(w_j)_{j\in J}]} \) the monomial order obtained by giving to \(w_j\) the weight \(w(w_j)=\widehat\nu (\eta_j)\in \Phi\) and considers the corresponding filtration by the weight, by passing to the associated graded rings on both sides one recovers the presentation \[ k[(W_j)_{j\in J}] \to \text{gr}_{\widehat\nu}\widehat R^{(\nu)},\qquad W_j \mapsto\overline \eta_j, \] of statement b) above for \((\widehat R^{(\nu)},\widehat\nu)\). The \((w_j)_{j\in J}\) (\(J\) is an ordinal smaller that \(\omega^r\) where \(r\) is the rank of \(\nu\)) are the coordinates proposed by Teissier.
The relations between the \(\overline \eta_j\) are binomial: \(\overline \eta_j ^m- \lambda_{m,n} \overline \eta_j ^n=0\), where \(m,n\) are multi-indices and \(\nu( \overline\eta_j ^m)=\nu( \overline\eta_j ^n)\). In this sense, \( \text{gr}_{\widehat\nu}\widehat R^{(\nu)}\) is toric. By the faithful flatness mentioned above, the relations between the \( \eta_j\) are of this type: \[ F_{mn}=\eta_j ^m- \lambda_{m,n} \overline \eta_j ^n+ \sum_{\widehat\nu (\eta_j ^s)>\widehat\nu( \eta_j ^m)} c_s \eta_j ^s=0. \] The specialization of these equations to their binomial initial form is the equational version of the specialization of \(\widehat R^{(\nu)}\) to the binomial variety.
This leads to
5.5. The abyssal phenomenon: Teissier conjectures a very precise result: since \(\widehat R^{(\nu)}\) is noetherian there should exist a finite set \(L\subset J\) such all the equations \(F_{mn}\) except finitely many of them serve only to express relations \(\eta_i= \sum_{l\in L} R_l \eta_l\). Thus they must contain linearly the variable \(\eta_i\) and cannot add new singularities. This really looks possible: \(\dim(m/m^2)\) is finite, so in the relations between the \(\eta_i\), almost all should appear linearly. The truth of this depends on an implicit function theorem in \(\widehat{k[(w_j)_{j\in J}]}\), not proved in the paper. If Teissier’s conjecture is true, we have a presentation of \(\widehat R^{(\nu)}\) as a quotient of \(\widehat{k[(w_j)_{j\in J}]}\) where almost all the equations contain a linear term and to uniformize \(\widehat\nu\) on \(\widehat R^{(\nu)}\) we have only to play with the others, which are a finite number of equations \(F_{mn}\) involving finitely variables, so that we are reduced to the simple case. The paper contains a scheme to deduce from this local uniformization of \(\nu\) on \(R\) if it is excellent.
This abyssal phenomenon must definitely be explored. The reviewer has the deep feeling that here is the key lemma, the finiteness lemma, that we have all sought for years and which will help also in the general desingularization problem.
Finally, the Jacobi-Perron algorithm, which was used in Zariski’s proof reappears here as the key step in the proof of a graded version of local uniformization, which is needed for the reduction to the simple case: The graded algebra \(\text{gr}_\nu R_\nu\) of the valuation ring is the union of a nested sequence of polynomial algebras \(P^{(h)}=k_\nu[x_1^{(h)},\ldots ,x_r^{(h)}]\) in \(r=\operatorname{rank}\nu\) variables, the maps \(P^{(h)}\to P^{(h+1)}\) sending each variable to a constant times a monomial in the new variables.
As happened many times in the history of this subject, the Guru showed us the right way…
For the entire collection see [Zbl 1021.00011].


14E15 Global theory and resolution of singularities (algebro-geometric aspects)
16W50 Graded rings and modules (associative rings and algebras)
13A18 Valuations and their generalizations for commutative rings
Full Text: arXiv