Herwig Hauser

Strong resolution of singularities in characteristic zero

Abstract. We present a concise proof for the existence and construction of a strong resolution of excellent schemes of finite type over a field of characteristic zero. Our proof is based on earlier work of Villamayor, Encinas-Villamayor and Bierstone-Milman. It proposes some substantial simplifications which may be helpful for a better understanding of how to prove Hironakas famous theorem on embedded resolution of singularities.

The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

This paper is a handyman’s manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups. Three objectives: Pleasant writing, easy reading, good understanding. One topic: How to prove resolution of singularities in characteristic zero. Statement to be proven (No-Tech): The solutions of a system of polynomial equations can be parametrized by the points of a manifold. Statement to be proven (Low-Tech): The zero-setX of finitely many real or complex polynomials in n variables admits a resolution of its singularities (we understand by singularities the points where X fails to be smooth). The resolution is a surjective differentiable map ε from a manifold X̃ to X which is almost everywhere a diffeomorphism, and which has in addition some nice properties (e.g., it is a composition of especially simple maps which can be explicitly constructed). Said differently, ε parametrizes the zero-set X (see Figure 1). Figure 1. Singular surface Ding-dong: The zero-set of the equation x + y = (1 − z)z in R can be parametrized by R via (s, t)→ (s(1− s) · cos t, s(1− s) · sin t, 1− s). The picture shows the intersection of the Ding-dong with a ball of radius 3. Received by the editors June 25, 2002, and, in revised form, December 3, 2002. 2000 Mathematics Subject Classification. Primary 14B05, 14E15, 32S05, 32S10, 32S45. Supported in part by FWF-Project P-15551 of the Austrian Ministry of Science. c ©2003 American Mathematical Society

On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for)

Assume that, in the near future, someone can prove resolution of singularities in arbitrary characteristic and dimension. Then one may want to know why the case of positive characteristic is so much harder than the classical characteristic zero case. Our intention here is to provide this piece of information for people who are not necessarily working in the field. A singularity of an algebraic variety in positive characteristic is called wild if the resolution invariant from characteristic zero, defined suitably without reference to hypersurfaces of maximal contact, increases under blowup when passing to the transformed singularity at a selected point of the exceptional divisor (a so called kangaroo point). This phenomenon represents one of the main obstructions for the still unsolved problem of resolution in positive characteristic. In the present article, we will try to understand it.

Affine varieties and lie algebras of vector fields

In this article, we associate to affine algebraic or local analytic varieties their tangent algebra. This is the Lie algebra of all vector fields on the ambient space which are tangent to the variety. Properties of the relation between varieties and tangent algebras are studied. Being the tangent algebra of some variety is shown to be equivalent to a purely Lie algebra theoretic property of subalgebras of the Lie algebra of all vector fields on the ambient space. This allows to prove that the isomorphism type of the variety is determinde by its tangent algebra.

Seventeen Obstacles for Resolution of Singularities

This is the first of a series of papers related to resolution of singularities. We present here examples which explain why many arguments and proofs work in special situations, say small dimension or zero characteristic, but fail in general. This exhibits in particular the delicacy of resolution of singularity for arbitrary excellent schemes. The examples were originally assembled for the author’s personal records. They might be of some interest to a larger audience, especially to readers for whom the flavour of resolution of singularities is concealed by technique.

A rank theorem for analytic maps between power series spaces

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Excellent Surfaces and Their Taut Resolution

Purpose of the present paper is to reveal the beauty and subtlety of resolution of singularities in the case of excellent two-dimensional schemes embedded in three-space and defined over an algebraically closed field of arbitrary characteristic. The proof of strong embedded resolution we describe here combines arguments and techniques of O. Zariski, H. Hironaka, S. Abhyankar and the author.

Today’s menu: Geometry and resolution of singular algebraic surfaces

The courses are Triviality, Tangency, Transversality, Symmetry, Simplicity, Singularity. These characteristic local plates serve as our invitation to algebraic surfaces and their resolution. Please take a seat.

Three power series techniques

The techniques and concepts we present are flags of regular schemes and their persistence under blow-up, the Gauss?Bruhat decomposition of the group of formal automorphisms of affine space, and coordinate-free initial ideals. All three are used to construct and study invariants for resolution of singularities.

Algebraic singularities have maximal reductive automorphism groups

Let X = O n /i be an analytic singularity where ṫ is an ideal of the C -algebra O n of germs of analytic functions on ( C n , 0). Let denote the maximal ideal of X and A = Aut X its group of automorphisms. An abstract subgroup equipped with the structure of an algebraic group is called algebraic subgroup of A if the natural representations of G on all “higher cotangent spaces” are rational. Let π be the representation of A on the first cotangent space and A 1 = π ( A ).