Algebraic Geometry Seminar
Fall 2010 — Tuesdays 3:304:30, LCB 323
Date  Speaker  Title — click for abstract (if available) 
August 31 
Tommaso de Fernex University of Utah 
The valuation space of an isolated normal singularity.
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 QGorenstein), we also obtain interesting applications to global geometry, and address questions of the following type:

September 7 
Jeremy Pecharich UC Irvine 
Deformation Quantization of Vector Bundles
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.

September 10 
Olivier Serman Université de Lille 1 
Local factoriality through products and quotients
(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 Zariskiopen
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.

September 28 
Aaron Bertram University of Utah 
Algebraic Geometry in Outer Space
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.

October 5 
Christopher Hacon University of Utah 
— 
October 19 
Rob Easton University of Utah 
Good quotients and good moduli spaces
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.

October 26 
Kevin Tucker University of Utah 
On the behavior of multiplier ideals and test ideals under
finite morphisms in positive characteristic
The multiplier ideal of a Qdivisor 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.

November 30 
José
González Univ. of Michigan 
Cox rings and pseudoeffective cones of projectivized toric vector bundles
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, HausenSuss, 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
[GonzalezHeringPayneSuss]. 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.

December 7 
Steffen Marcus Brown University 
Polynomial Families of Tautological Classes on the Moduli Space
of Curves
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. 
December 14 
David Swinarski University of Georgia 
Vector bundles of conformal blocks on
M_{0,n} for sl_{2} and sl_{n}
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 sl_{2} 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.

Archive of previous seminars.
This web page is maintained by Jimmy Dillies and Roi Docampo Álvarez.