Steven Dale Cutkosky

Resolution of singularities

Introduction Non-singularity and resolution of singularities Curve singularities Resolution type theorems Surface singularities Resolution of singularities in characteristic zero Resolution of surfaces in positive characteristic Local uniformization and resolution of surfaces Ramification of valuations and simultaneous resolution Smoothness and non-singularity Bibliography Index.

Resolution of singularities for 3-folds in positive characteristic

A concise, complete proof of resolution of singularities of 3-folds in positive characteristic

Monomialization of morphisms from 3-folds to surfaces

>5

Ramification of valuations

is given. Abhyankar first proved this theorem in 1966.

On unique and almost unique factorization of complete ideals II

1. Introduction.- 2. Local Monomialization.- 3. Monomialization of Morphisms in Low Dimensions.- 4. An Overview of the Proof of Monomialization of Morphisms from 3 Folds to Surfaces.- 5. Notations.- 6. The Invariant v.- 7. The Invariant v under Quadratic Transforms.- 8. Permissible Monoidal Transforms Centered at Curves.- 9. Power Series in 2 Variables.- 10. Ar(X).- 11.Reduction of v in a Special Case.- 12. Reduction of v in a Second Special Case.- 13. Resolution 1.- 14. Resolution 2.- 15. Resolution 3.- 16. Resolution 4.- 17. Proof of the main Theorem.- 18. Monomialization.- 19. Toroidalization.- 20. Glossary of Notations and definitions.- References.

Asymptotic Behavior of the Length of Local Cohomology

Abstract In this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields. In characteristic zero and arbitrary dimension, we obtain the strongest possible generalization of the classical ramification theory of local Dedekind domains. We further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic. We prove that local monomialization and simultaneous resolution hold under very mild assumptions, and give pathological examples.

Poincaré series of resolutions of surface singularities

The theory of complete ideals was begun by Zariski in 1938 [Z 1], and further developed in Appendix 5 of [ZS]. Zariski developed a theory of factorization of complete ideals in regular two-dimensional local rings. In this paper we develop the theory of factorization of complete ideals in complete normal two-dimensional local rings. Let (R, m) be a complete normal two-dimensional local ring, and let re(R) be the semi-group of complete m-primary ideals of R. We show how the divisor class group of R, CI(R), is related to factorization in re(R). We generalize results obtained in [C2] where it is assumed that R/m is algebraically closed. Assume that R is any regular local ring of dimension 2. Zariski proved that any complete ideal I of R has unique factorization into products of simple complete ideals. The distinct simple complete ideals occuring in the factorization of I are in 1 1 correspondence with the irreducible components of the exceptional divisor of the blowing up of I. This shows that any proper birational morphism X ~spec(R) with X normal can be decomposed into a sequence of blowups with irreducible exceptional divisors. Now assume that R is a complete normal local domain of dimension 2. It is natural to ask if the condition that m(R) has unique factorization is equivalent to R being a UFD. We show (Theorem 5) that if re(R) has unique factorization then R is a UFD. We also give an example (Example 1) to show that the converse does not hold. However, Lipman has shown in [L] that if R is a UFD, and if R/m is algebraically closed, then re(R) has unique factorization. Recall that R is a UFD if and only if CI(R)=0. G6hner [G] introduced the notion of a semigroup being semi-factorial. This definition is reproduced in definition 3 of this paper. We prove G6hners conjecture (2 3 of Theorem 4) that re(R) is semifactorial if and only if CI(R) is a torsion group. As a corollary (Corollary to Theorem 4) we obtain that if Rim is algebraically closed of characteristic zero, then re(R) is semi-factorial if and only if R has a rational singularity. Another corollary is that CI(R) is torsion if and only if R satisfies the condition (N) of Muhly and Sakuma [-MS] (1r of Theorem 4). R satisfies condition (N) if any proper birational morphism X ~ spec(R) factors as a sequence of blowups of height 2 ideals with irreducible exceptional divisors.

Valuation semigroups of two-dimensional local rings

Let kbeafieldofcharacteristic 0, R = k(x1,..., xd)beapolynomialring,and mitsmaximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. Let �(M) denote the length of an R- module M. In this paper, we show that lim n!1 � H 0 m(R/I n ) � nd = lim n!1 � Ext d R/I n , R(−d) � � nd

On Fano 3-folds

Let X → spec(R) be a resolution of singularities of a normal surface singularity spec(R), with integral exceptional divisors E 1 ,..., E r . We consider the Poincare series g = h(n)t n , where h(n) = (R/Γ(X, O x (-n 1 E - 1 - ... - n r E r )). We show that if R/m has characteristic zero and Pic 0 (X) is a semi-abelian variety, then the Poincare series g is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.

Counterexamples to local monomialization in positive characteristic

We consider the question of when a semigroup is the semigroup of a valuation dominating a two dimensional noetherian local domain, giving some surprising examples. We give a necessary and sufficient condition for the pair of a semigroup S and a field extension L/k to be the semigroup and residue field of a valaution dominating a regular local ring R of dimension two with residue field k, generalizing the theorem of Spivakovsky for the case when there is no residue field extension.