Algebraic Geometry Seminar

The Zariski space of a normal variety X encodes all resolutions of X. In joint work with Charles Favre and Sebastien Boucksom, we study positivity properties of divisors on the Zariski space of a normal variety X. In the case X has an isolated singularity, we concurrently work on the full space of valuations of rank one centered at the singular point and study properties of various functions on it.

The motivation of our work comes from an old paper of Wahl, where he defines a "characteristic number" of the surface singularity -- an invariant that vanishes if and only if the singularity is log canonical. Our goal is to extend Wahl's result to all dimensions. By working in full generality (especially, without assuming X to be Q-Gorenstein), we also obtain interesting applications to global geometry, and address questions of the following type:

• Which projective varieties admit polarized finite endomorphisms of degree > 1?
• Is the property of being log-Fano preserved by finite morphisms of projective varieties?

Let X be an algebraic Poisson variety and Y a smooth coisotropic subvariety with a vector bundle E supported on Y. We give a criterion for when E admits a first/second order deformation as a left module over the first/second order deformation quantization of O_X. We will then show noncommutative module deformations are controlled by a curved dg Lie algebra which reduces to the classical relative Hochschild complex when the Poisson structure is trivial. Part of this work is joint with Vladimir Baranovsky and Victor Ginzburg.

(Q-)factoriality of local rings defines a fairly nice kind of singularities. However, to some extent, this notion is not so well behaved. In particular, it is not local in the étale topology. In this talk we show that it is a Zariski-open property. We investigate then how local factoriality is preserved by taking products. Unsurprisingly, factoriality of GIT quotients is far more involved. If time permits, we will explain how a very basic result describing divisor class groups of local rings in a quotient easily leads to non trivial information about the singular locus of some moduli spaces of bundles on curves.

Classical Riemann surface theory has an interesting analogue for metric graphs (connected graphs whose edges are assigned lengths) in which the complex numbers are replaced by the tropical numbers and rational functions are replaced by piecewise linear functions with integer slopes. I'll explore some (hopefully interesting) metric graph versions of classical constructions for Riemann surfaces and talk a little about how outer space is and isn't like the moduli space of Riemann surfaces from this point of view.

Algebraic stacks are undeniably technical objects. However, once one comes to terms with their abstract nature (or simply accepts them as black boxes), they become incredibly useful tools. This applies not only to modern (and otherwise intractable) problems, but also to classical questions, especially those in equivariant geometry. I will summarize one such example, in which the language of algebraic stacks can be used to quickly reprove (and even generalize) a statement on the existence of good quotients.

The multiplier ideal of a Q-divisor on a complex algebraic variety is a fundamental object in the study of higher dimensional birational geometry. However, the behavior of multiplier ideals in positive characteristic can be quite enigmatic. In many cases, a related invariant called the test ideal displays preferable behavior. In this talk, we will review the relationship between the multiplier ideal and test ideal in positive characteristic. Furthermore, we will describe (and contrast) transformation rules for each of these invariants under an arbitrary (i.e. not necessarily separable) finite morphism. This is joint work with Karl Schwede.

Projectivized toric vector bundles are a large class of rational varieties that share some of the finiteness properties of toric varieties and other Mori dream spaces. Hering, Mustata and Payne proved that the Mori cones of these varieties are polyhedral and asked whether their Cox rings are indeed finitely generated. In this talk we give a complete answer to this question. There are now several proofs of a positive answer in the rank two case [Knop, Hausen-Suss, Gonzalez]. For any rank greater than two we present projectivized toric vector bundles for which the Cox ring and the pseudoeffective cones can be identified with those of the projective space blown up at a finite set of points of our choice [Gonzalez-Hering-Payne-Suss]. This yields many new examples of Mori dream spaces, as well as examples of projectivized toric vector bundles where the pseudoeffective cone is not polyhedral and the Cox ring is not finitely generated.

The tautological ring is a heavily studied subring of the intersection ring of the moduli space of curves. Simply stated, it is just large enough to contain all the known Chow classes admitting some geometric construction. In this talk, I will describe natural families of tautological classes which arise by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized projective line. 'Relative' in this case means our maps have prescribed ramification over zero and infinity given by partitions of the degree. A theorem of Vakil shows the families are polynomial in the parts of the partitions. I will discuss our approach to computing these polynomials, involving both virtual localization as well as some surprising combinatorics.
This is joint work with Renzo Cavalieri.

The WZW model of conformal field theory yields a vector bundle on the moduli space of pointed curves M_0,n depending on a choice of a Lie algebra, a level, and a set of n weights in the corresponding Weyl alcove. Recently, Fakhruddin gave formulas for the Chern classes of these bundles when g=0 and showed they are globally generated. I will discuss recent joint work with Alexeev, Arap, Giansiracusa, Gibney, and Stankewicz on these bundles for sl₂ and sl_n. We show that some of these divisors associated to these bundles are extremal in the nef cone and identify the images of the corresponding linear systems.