In AGNES - MIT 2011 Caporaso described a
compelling analogy between the Deligne-Mumford-Knudsen
moduli space and the space of stable tropical curves. We
show how to realize this analogy as the outcome of the
extended tropicalization described by Payne in the same
AGNES workshop.This is joint work in progress with Lucia Caporaso and Sam Payne.
It has been known for quite a while that among the infinitely many toroidal compactifications of the moduli space of polarized abelianvarieties A, the compactification corresponding to the 2nd Voronoi fan has a moduli meaning, and that the Torelli map to it is regular and has a moduli interpretation._{g}In this talk, I will discuss whether OTHER toroidal compactifications of A and the corresponding extended Torelli maps can be given a geometric interpretation. Time permitting, I will discuss similar questions for moduli spaces of K3 surfaces._{g}Parts of this story are based on joint works with Brunyate, and with Livingston-Tenini et al.
I will discuss some aspects of the geometry of spaces of rational curves
on Fano hypersurfaces (or more generally complete intersections)
in projective space and Grassmannians. Some of the basic properties of these
spaces are still unknown. The focus of my talk will be on some questions
regarding the dimension and irreducibility.
This talk is partly based on joint work with N. Mohan Kumar.
Deligne observed a long time ago that the category of split real Hodge
structures is equivalent to the category of representations of a
certain rank 2 torus. Later he described the Tannakian Galois group
of the category of all real mixed Hodge structures as a certain
extension of this torus. I will discuss this and joint work with
Greg Pearlstein where we spell out a similar picture for
degenerations of real mixed Hodge structures.
A quadratic differential (resp. abelian
differential) defines a flat (resp. very flat) structure
on a Riemann surface, such that it can be visualized as a
plane polygon. Varying the shape of the polygon induces a
natural SL(2)-action on their moduli space. In this talk
I will describe the geometry of this moduli space as well
as a closed SL(2)-orbit, called Teichmueller curve. Using
a special Hurwitz curve as example, I will illustrate a
beautiful interplay between polygon billiards, counting
branched covers and the intersection theory on moduli
space. The numerical property of Teichmueller curves
will be discussed in case of low genus (joint with Martin
Moeller) and high genus (little is known).
For two centuries, the question of the rationality of cubic hypersurfaces has played a pivotal role in algebraic geometry. The discovery of the irrationality of cubic plane curves and the rationality of cubic surfaces, and most recently the proof by Clemens and Griffiths that smooth cubic threefolds are irrational, have all been milestones in the subject.
At issue now is whether smooth cubic fourfolds are rational. While the issue is still very much open, it seems likely that it will illuminate another question: whether rationality is an open condition in families of smooth projective varieties, and whether it's closed (both are unknown in general).
In this talk I'll give an overview of the state of our knowledge about cubic fourfolds, the conjectured answer to the question of rationality and why we might believe it's true.
Let X be a smooth projective variety with a numerically trivial canonical line-bundle. The ample and movable cones of X can have infinitely many linear as well as circular boundary faces, and are often quite complicated. Morrison's cone conjectures state, roughly, that the ample cone is simple modulo the action of the automorphism group of X, and the movable cone is simple modulo the action of the group Bir(X) of birational automorphisms of X.A version of the movable cone conjecture in the title is derived as a corollary of the Global Torelli Theorem for irreducible holomorphic symplectic manifolds. As a consequence it is shown that for each non-zero integer d there are only finitely many Bir(X)-orbits of complete linear systems, which contain a reduced and irreducible divisor of Beauville-Bogomolov degree d. A similar finiteness result holds in degree zero as well.
The non-nef locus of a big divisor is a subset
that measures how far the divisor is from being nef (when
the divisor admits a Zariski decomposition
on some birational model, the non-nef locus is the image
of the support of the negative part in this
decomposition). I will discuss a description of this
locus in positive characteristic, using test ideals. |

Stony Brook 2011 >