## AGNES - Stony Brook 2011October 28-30, 2011 Stony Brook, New York |

**Eyal Markman**(University of Massachusetts Amherst)

Morrison's movable cone conjecture for irreducible holomorphic symplectic varieties

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.

**Patrick Brosnan**(University of Maryland and University of British Columbia)

Degenerations of Mixed Hodge Structure from a Tannakian viewpoint

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.

**Valery Alexeev**(University of Georgia)

Geometric meaning of toroidal compactifications of moduli spaces

It has been known for quite a while that among the infinitely many toroidal compactifications of the moduli space of polarized abelian varieties*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 Ag 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.

Parts of this story are based on joint works with Brunyate, and with Livingston-Tenini et al.

**Joe Harris**(Harvard University)

Rationality of Cubic Fourfolds

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.

**Roya Beheshti**(Washington University)

Spaces of rational curves on general hypersurfaces

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.

**Dawei Chen**(Boston College)

Flat surfaces, moduli of differentials and Teichmueller curves

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).

**Mircea Mustaţă**(University of Michigan, Ann Arbor)

The non-nef locus in positive characteristic

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.

**Dan Abramovich**(Brown University)

The tropicalization of moduli space

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.