Algebraic Geometry Seminar Spring 2013
The seminar meets on Fridays at 2:25 pm in Van Vleck B219.
The schedule for the previous semester is here.
|January 25||Anatoly Libgober (UIC)||Albanese varieties of cyclic covers of plane, abelian varieties of CM type and orbifold pencils||Laurentiu|
|February 1||Laurentiu Maxim (University of Wisconsin-Madison)||Intersection spaces, perverse sheaves and type IIB string theory||local|
|March 1||Alexander Polishchuk (University of Oregon)||Lefschetz theorems for dg-categories with applications to matrix factorizations||Dima|
|March 15||Xue Hang (Columbia)||On the height of a canonical point in the Jacobian of a genus four curve||Tonghai|
|April 12||Nick Rozenblyum (Northwestern)||B-model using derived algebraic geometry||Andrei|
|April 17, 1:20 PM, Room 901||Xavier Gomez-Mont (CIMAT, Guanajuato, Mexico)||Signatures on the Primitive Parts of the Real Jacobian Algebra.||Laurentiu|
|April 26||Jack Huizenga (University of Illinois-Chicago)||Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles||Melanie|
|May 3||Vladimir Baranovsky (University of California - Irvine)||Poisson deformations of coherent modules||Dima|
|May 10||Yu-jong Tzeng (Harvard University)||Counting curves with higher singularities on surfaces||Melanie|
Albanese varieties of cyclic covers of plane, abelian varieties of CM type and orbifold pencils
I'll describe the relation between Alexander modules of plane algebraic curves and maps of their complements onto orbifolds. A key step is a description of the Albanese variety of cyclic covers of the plane in terms of abelian varieties of CM type.
Intersection spaces, perverse sheaves and type IIB string theory
The method of intersection spaces associates rational Poincare complexes to singular stratified spaces. For a complex projective hypersurface with only isolated singularities, we show that the cohomology of the associated intersection space is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. We will discuss properties of the intersection space complex, such as self-duality, its betti numbers and mixed Hodge structures on its hypercohomology groups. This is joint work with Banagl and Budur.
Lefschetz theorems for dg-categories with applications to matrix factorizations
I will describe versions of Lefschetz type formulas in the context of dg-categories. I will consider the case of the dg-category of matrix factorizations of an isolated hypersurface singularity and will show explicit calculations of the ingredients of the Lefschetz formula in this case.
On the height of a canonical point in the Jacobian of a genus four curve
In this talk, we construct a quadratic point in the Jacobian of a non-hyperelliptic curve of genus four over a global field. We then compute the Neron--Tate height of this point in terms of the self-intersection of the admissible dualizing sheaf and some canonically defined local invariants. We show that the height of this point satisfies the Northcott property. We also give some estimates of the local invariants that appear in the height computation. When the reduction of the curve is simple, we compute explicitly the local invariants.
B-model using derived algebraic geometry
The B-model is a 2D topological quantum field theory, which gives operations, parametrized by the moduli space of pointed curves, on the Hodge cohomology of a Calabi-Yau variety. I will describe a geometric construction of these operations, using integral kernels in derived algebraic geometry. This construction is similar in spirit to Feynman integration in quantum field theory.
Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles
The Hilbert scheme of n points in the projective plane parameterizes zero-dimensional subschemes of length n. An interesting problem is to describe the birational geometry of this space, and give modular interpretations for its various birational models. A first step in this program is to determine the cone of effective divisors on the Hilbert scheme.
We show the sections of many stable vector bundles satisfy a natural interpolation condition, and that these bundles always give rise to the edge of the effective cone. To do this, we give a generalization of Gaeta’s theorem on the resolution of the ideal sheaf of a general collection of n points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that general ideal sheaves are always destabilized by exceptional bundles.
Poisson deformations of coherent modules
Suppose that F is a coherent sheaf on a smooth variety X, and that the structure sheaf of X admits a deformation quantization. We discuss the problem of deforming F to a module over the quantized algebra. We state some necessary conditions on F and consider a special case when they are sufficient.
Counting curves with higher singularities on surfaces
A famous problem in classical algebraic geometry is how many r-nodal curves are there in a linear system |L| on an algebraic surface S. If the line bundle L is sufficiently ample, Gottsche conjectured that the number of r-nodal curves is a universal polynomial of Chern numbers of L and S for any r. This conjecture was proven independently by Tzeng and Kool-Shende-Thomas. In this talk we will generalize Gottsche's conjecture and show the numbers of curves with any number of arbitrary isolated singularity on surfaces are also given by universal polynomials. Moreover these polynomials can be combined to form a huge generating series and we will discuss its properties.