Summer School 2012

Clay Mathematics Institute Summer School 2012

The Resolution of Singular Algebraic Varieties

June 3 - June 30, 2012
Obergurgl, Tyrolean Alps, Austria


Schedule: Week 1, Week 2, Week 3, Week 4



The resolution of singularities is one of the major topics in algebraic geometry. Due to its difficulty and complexity, as well as certain historical reasons, research to date in the field has been pursued by a relatively small group of mathematicians. However the field has begun a renaissance over the last twenty years, boosted by many small conferences and schools, with the discovery of more conceptual proofs of the characteristic zero case, as well as several brilliant attempts at the still unresolved prime characteristic case. This school will consist of three weeks of foundational courses supplemented by exercise and problem sessions, designed to provide graduate students and young mathematicians with a comprehensive framework for research in this field. The fourth week will consist of mini-courses with selected experts, aimed at providing participants with state of the art techniques, as well as a survey of some of the main open problems and the most promising approaches now under investigation. The Obergurgl Center is situated at 2000m above sea level in the Tyrolean Alps, two hours from Innsbruck (by bus) and four hours from Munich (by train and bus), both with international airports. Facilities will be provided for lectures, meals and lodging at the center.

Foundational Courses

Blowups, centers, exceptional divisors, strict, weak, controlled and total transforms, upper semicontinuous functions, stratifications and filtrations, maximal contact, osculating hypersurfaces, descent in dimension, coefficient ideals, retractions and projections, transversality and normal crossings, flags, Cartesian induction, mobiles, local coordinates, automorphism groups, Newton polyhedra, local resolution, proof of resolution in zero characteristic, local uniformization, Nash modification, normalization, alterations, ...

Course Description

The proof of existence of resolution of singularities of algebraic varieties in characteristic zero can be divided into two parts. First, there is an algebraic part, providing necessary constructions such as blowups of manifolds along submanifolds, differential closure, transforms along blowup, descent in dimension, transversality conditions, and properties of these constructions. Second, there is a combinatorial part that consists in the setup of a tricky form of double induction taking various side conditions into account. These two parts can be cleanly separated: once the properties of the algebraic constructions are stated in an axiomatic way, it is no more necessary to do any algebra in the induction proof. The goal of this lecture will be two-fold. First, we want to introduce all the algebraic constructions mentioned above, prove their properties, and provide algorithms that carry them out effectively. For this purpose, we will use the computer algebra system MAGMA; this system has built-in functions that will be convenient (variable elimination, syzygy computation). The participants are expected to do various exercises. Second, we will translate the axiomatic description of the properties of the algebraic constructions (blowups etc.) into rules of a combinatorial game between two players. We will prove that the second player has a winning strategy; this implies the existence of resolutions of singularities.

Prerequisites: For the algebraic part, participants are assumed to be familiar with basic concepts in algebraic geometry: schemes, projective morphisms, syzygies. The combinatorial part requires no background.

Course Description

Regular varieties. Monoidal transformations on regular varieties, Higher differential operators. The notions of order of an ideal and of multiplicity for an embedded hypersurface. Rees algebras and differential algebras. Integral closure of algebras and their monoidal transforms. Special features of positive characteristic. Elimination algebras. Application to resolution of singularities.

Course Decription

Mini-Courses

The fourth week will consist of mini-courses aimed at a much higher level.

We will present Rees algebras as a tool to address different problems in resolution of singularities over perfect fields. In particular, we will focus on the problem of globalization of local invariants in characteristic zero, and study questions related to multiplicity in positive characteristic.

We give a survey of what is known in the subject, mostly presenting results and methods of Abhyankar and Hironaka. The first case considered is that of Artin Schreier extensions in dimension two, which already illustrate fundamental obstructions to resolution. From here we move to a discussion of how an algorithm for resolution of Artin Schreier extensions extends to a proof of embedded resolution of singularities in dimension two. Then we present a proof of resolution of 3-folds. We will also discuss some recent results in the area, and open problems.

Given a toric variety X it is well-known that by refinement of the fan defining X one may obtain a resolution of singularities. This gives a toric morphism. In this talk we will focus in a different direction, we will use binomials defining the toric variety instead of fans. The goal is to produce a sequence of combinatorial blowing ups such that the final transform is an embedded resolution. Every center of blowing up is computed using an upper-semicontinuous function, in a similar way as in the general case, but the coordinates are much simpler. This algorithm is valid for toric varieties over a perfect field of arbitrary characteristic.

In this series of talks, I will expose a recent result in collaboration with Michael Mcquillan on resolution of singularities for one-dimensional complex foliations in three dimensional ambient spaces. A special emphasis will be put on the main differences and difficulties with respect to the desingularization of singular varieties.

We will study higher versions of Nash blowup. These blowups provide a possibility to resolve singularities in a single step. It works only in limited cases. However the obtained desingularization has a nice description as a parameter space of geometric objects. We will define these blowups, study basic properties and survery known results. If time permits, also the computational aspect will be discussed.

Scientific Committee

Lecturers

Arrival Information

Abstracts of Talks

Foundational Courses

Mini-Courses