Algebraic Geometry Seminar Fall 2010

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The seminar meets on Fridays at 2:25 pm in Van Vleck B305.

A preliminary schedule for the spring semester is here.

Fall 2010

date speaker title host(s)
sept 24 Xinyi Yuan (Harvard) Calabi-Yau theorem and algebraic dynamics Tonghai
oct 1 Dawei Chen (UIC) Geometry of Teichmuller curves Andrei
oct 8 Tom Church (U Chicago) Representation stability, homological stability, and configurations over finite fields Jordan
oct 15 Conan Leung (CUHK) The quantum cohomology of G/P Andrei
oct 22 Zhiwei Yun (Berkeley) Springer representation and Hitchin fibration Shamgar
oct 23-24 (Sat/Sun) Midwest Graduate Algebraic Geometry Conference Conference Website UW Math
oct 29 Christian Schnell (UIC) Derived equivalence and the Picard variety Laurentiu
nov 5 Daniel Erman (Stanford) Sextic covers and Gale duality Jordan
nov 12 Luca Scala (Chicago) Symmetric powers of tautological bundles on Hilbert schemes of points on a surface Andrei
nov 19 Alina Marian (UIC) The moduli space of stable quotients Andrei
dec 3 Matt Satriano (UMich) Stacky resolutions of singular schemes David Brown
dec 10 Izzet Coskun (UIC) Restriction varieties and geometric branching rules Andrei

Abstracts

Xinyi Yuan Calabi-Yau theorem and algebraic dynamics

The uniqueness part of the Calabi-Yau theorem asserts that the Monge-Ampere measure of a (complex) positive hermitian line bundles determines the hermitian metric up to constant. Here we introduce a p-adic analogue of the theorem. Combinning with the equidistribution theory, we obtain the rigidity of preperiodic points on algebraic dynamical systems.


Dawei Chen Geometry of Teichmuller curves

We study Teichmuller curves parameterizing square-tiled surfaces (i.e. covers of elliptic curves with a unique branch point).

The results can be applied to the following problems in algebraic geometry and complex dynamics: (a) construct rigid curves on the moduli space of pointed rational curves; (b) bound the effective cone of the moduli space of genus g curves; (c) verify the invariance of Siegel-Veech constants; (d) calculate the Lyapunov exponents of the Hodge bundle.


Tom Church Representation stability, homological stability, and configurations over finite fields

Homological stability is a remarkable phenomenon where for certain sequences X_n of groups or spaces -- for example SL(n,Z), the braid group B_n, or the moduli space M_n of genus n curves -- it turns out that the homology group H_i(X_n) does not depend on n once n is large enough. However, there are many natural analogous sequences, from pure braid groups to congruence groups to Torelli groups, for which homological stability fails horribly. In these cases the rank of H_i(X_n) blows up to infinity, and in the latter two cases almost nothing is known about H_i(X_n); indeed it's possible there is no nice "closed form" for the answers.

While doing some homology computations for the Torelli group, we found what looked like the shadow of an overarching pattern. In order to explain it and to formulate a specific conjecture, we came up with a notion of "stability of a sequence of representations of groups". This makes it possible to meaningfully talk about "the stable homology of the pure braid group" or "the stable homology of the Torelli group" even though homological stability fails. We have proved that many important sequences are representation-stable in this sense, including the homology of pure braid groups, configuration spaces of manifolds, and certain Malcev Lie algebras, as well as sequences not arising from homology. For other cases, including Torelli groups and congruence subgroups, this notion provides a natural source of analogies and conjectures. In this talk I will explain our broad picture via some of its many instances, focusing on pure braid groups and configuration spaces. This work is joint with Benson Farb. One striking application is a surprisingly strong connection between representation stability for pure braid groups and counting problems for polynomials over finite fields, joint with Jordan Ellenberg and Benson Farb.


Zhiwei Yun Springer representation and Hitchin fibration

Classical Springer representation is the action of the Weyl group on the cohomology of certain subvarieties of the flag variety. I will construct a global analogue of this action, namely, an action of the graded double affine Hecke algebra on the cohomology of parabolic Hitchin fibers. Examples in SL(2) will be described in details. This construction is motivated by Ngo's proof of the fundamental lemma, and has applications to the harmonic analysis on p-adic groups.


Christian Schnell Derived equivalence and the Picard variety

I will explain a result, joint with Mihnea Popa, saying that if two smooth projective varieties have equivalent derived categories of coherent sheaves, then their Picard varieties are isogeneous. In particular the number of independent holomorphic one-forms is a derived invariant. A consequence of this is that derived equivalent threefolds have the same Hodge numbers.


Dan Erman Sextic covers and Gale duality

The moduli space of n to 1 covers is well understood for n at most 5, and it turns that these moduli spaces are best understood in terms of a rather concrete question: when can the ideal of n points in projective space be generated by the minors of a matrix of linear forms? I will first explain some of what was previously known about the moduli of degree n covers, and then I will discuss some recent progress on the case of sextic covers. In particular, by illustrating a connection with Gale duality, we identify local and global obstructions to extending previous structural theorems to the sextic case. This is joint work with Melanie Matchett Wood.


Luca Scala Symmetric powers of tautological bundles on Hilbert schemes of points on a surface

By techniques by Danila and Le Potier, the comprehension of global sections of certain determinant line bundles on moduli spaces of sheaves over the projective plane can be reduced to the understanding of the cohomology of symmetric powers of some tautological line bundles on Hilbert schemes of points on P_2. We will discuss a work in progress on these symmetric powers, in order to understand their global sections, and some ideas to get their higher cohomology.


Alina Marian The moduli space of stable quotients

I will discuss a moduli space of stable sheaf quotients of the rank n trivial sheaf on curves. It gives a compactification of the moduli of maps from smooth curves to Grassmann varieties, distinct from the Kontsevich-Manin space. The construction connects with Gromov-Witten theory on the one hand, and leads to new relations in the tautological ring of the moduli space of curves on the other. This is based on joint work with Dragos Oprea and Rahul Pandharipande.


Izzet Coskun Restriction varieties and geometric branching rules

In representation theory, a branching rule describes the decomposition of the restriction of an irreducible representation to a subgroup. Let $i: F' \rightarrow F$ be the inclusion of a homogeneous variety in another homogeneous variety. The geometric analogue of the branching problem asks to calculate the induced map in cohomology in terms of the Schubert bases of $F$ and $F'$. In this talk, I will give a positive, geometric rule for computing the branching coefficients for the inclusion of an orthogonal flag variety in a Type-A flag variety. The geometric rule has many applications including to the restrictions of representations of $SL(n)$ to $SO(n)$, to the study of the moduli spaces of rank 2 vector bundles on hyperelliptic curves and to presentations of the cohomology ring of orthogonal flag varieties.