Algebraic Geometry Seminar

Consider a branched covering map from an irreducible projective variety X to a smooth variety Y. In 1980, Gaffney and Lazarsfeld extended the notion of branched locus to “higher ramification loci” which stratifies the branched locus, and they proved that when Y is a projective space, there are anticipated codimension bounds for the loci. I will explain their method with some background, and show that this result is also true when Y is a projective homogeneous space with Picard number 1.

I will discuss recent work of Bhargav Bhatt, myself and Shunsuke Takagi relating several open problems and generalizing work of Mustata and Srinivas. First: whether a smooth complex variety is ordinary after reduction to characteristic $p > 0$ for infinitely many $p$. Second: that multiplier ideals reduce to test ideals for infinitely many $p$ (regardless of coefficients). Finally, whether complex varieties with Du Bois singularities have $F$-injective singularities after reduction to infinitely many $p > 0$.

We抣l discuss various constraints on rational curves homeomorphic to their normalizations in the complex projective plane, and consider the question of whether every such curve is necessarily equivalent to a line, under a birational automorphism of the plane.

We construct a global geometric model for heterotic - F-theory duality, that is, a Calabi-Yau threefold with two flat E_8 bundles with SU(5)-symmetry (heterotic side) and a Calabi-Yau fourfold with SU(5) root symmetry (F-theory side), both equipped with compatible involutions that, in the respective quotients, break the SU(5)-symmetry to the SU(3)xSU(2)xU(1)-symmetry of the physics 'standard model.' The duality is realized by an equivariant semi-stable degeneration of the Calabi-Yau fourfold to the union of two components, each of which is a so-called DP9-bundle containing a common Calabi-Yau threefold (along which the two components are joined). A result of Friedman-Morgan-Witten, gives a dictionary between flat E_8 bundles and DP9-bundles. (NB: The terminology DP9 stands for 'Del Pezzo 9-surface' which the string theorists use for something that is, in fact, not a Del Pezzo surface but what you get when you blow up the one-point base locus of |-K| in a Del Pezzo surface with (-K)^2 = 1.) This is joint with S. Raby and T. Pantev.

In this talk I will present results demonstrating that derived equivalence between varieties over finite fields that are either abelian or surfaces implies equality of zeta functions.

I will describe an elementary construction of invariants of higher genus curves which are analogous to the j-invariant of an elliptic curve. Then I will discuss how the j-invariant and its higher genus analogs appear in the study of certain A-infinity algebras associated with algebraic curves. I will also explain how a similar construction in the case of genus 1 curves with n marked points leads to a non-standard modular compactification of M_{1,n} studied earlier by Smyth (this is a joint work with Yanki Lekili).

I will present the Landau-Ginzburg mirror symmetry conjecture and its proof for all invertible singularities. The proof includes a reconstruction theorem for the A-model (FJRW theory) and the B-model (Saito-Givental theory) which states that the all-genus potential is determined by certain terms of the the genus-zero four-point function. This work is joint with Weiqiang He, Si Li, and Yefeng Shen.

Given an algebraic curve with marked points and integer weights on the markings, we can consider the corresponding divisor. This defines an Abel--Jacobi map from the moduli space of marked curves of compact type to the universal Jacobian variety. Pulling back relations from the universal Jacobian for various values of the weights gives a plethora of tautological relations on the space of curves of compact type. These relations include those discovered by Faber, Getzler, Belorousski and Pandharipande, and Tavakol, and this method was also used by Hain to compute the double ramification cycle. I will talk about the extension of the Abel--Jacobi map to a larger moduli space of marked curves, the corresponding extended formula for the double ramification cycle, and Pixton's conjectures about the extensions of these relations to the entire Deligne-Mumford compactification.

On a nonnormal variety, the limit of a family of line bundles is not always a line bundle. What is the limit? I will present an answer to this question and give some applications. Time permitting, I will discuss connections with Néron models, autoduality, and recent work of R. Hartshorne and C. Polini.

Divisors exhibit a close relationship between numerical "positivity" and asymptotic invariants. I will discuss the analogous principle for cycle classes of arbitrary codimension.

The cone of effective divisors contains important information about the birational geometry of a variety. In this talk I will give an introduction to this subject, with a focus on the case when the ambient variety is the moduli space of curves. If time permits, I will also talk about effective cycles of higher codimension.

Strange duality is a conjectural perfect pairing between sections of determinant (or theta) line bundles on moduli spaces of sheaves. Marian and Oprea resolved this conjecture in the affirmative for curves by constructing a finite Quot scheme whose points gave a basis for the image of the strange duality map. We attempt to use this strategy on surfaces, the most accessible case being del Pezzo surfaces. Some of the Quot schemes we construct will be Quot schemes in a tilted category. I will discuss some interesting computations and partial results that support our conjecture.

There is a close connection between the geometry of a smooth curve, and the geometry of its Jacobian. In this talk I will discuss joint work with J. Kass and F. Viviani where we investigate the connection between the geometry of a stable curve and its compactified Jacobians, constructed by Caporaso, Oda-Seshadri, Pandharipande, and Simpson. Specifically, we describe the singularities of the spaces, as well as singularities of their theta divisors in the integral case. Applications to the birational geometry of universal compactified Jacobians will also be discussed.

Abstract: Let A be an affine variety inside a complex N dimensional vector space which has an isolated singularity at the origin. The intersection of A with a very small sphere turns out to be a manifold called the link of A. The link has a natural hyperplane distribution called a contact structure. If the singularity is numerically \Q Gorenstein then we can assign an invariant of our singularity called the minimal discrepancy. We relate the minimal discrepancy with the contact geometry of our link. As a result we show that if the link of A is contactomorphic to the link of \C^3 and A is normal then A is smooth at 0. This generalizes a Theorem by Mumford in dimension 2.