Algebraic Geometry Seminar

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties of Segre products that are Gorenstein. Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous results for the tangential variety.

The deformation of the Hilbert scheme of points on the projective plane is studied by Hitchin, Nevins and Stafford via different approaches. I will introduce these constructions, and talk about my recent results on the minimal model program of the deformation of Hilb P2.

Tautological classes form a subring of the Chow ring of the moduli space of curves. There is an explicit set of generators but the set of relations remains unknown. In 2012 Aaron Pixton gave an explicit set of conjectural relations and gave a lot of evidence that these are actually all the relations. I want to compare two proofs of the fact that the conjectural relations are actual relations.

Tautological classes are certain elements of the cohomology or Chow ring of the moduli space of curves that are important in Gromov-Witten theory. We describe a method for deriving relations between these classes by studying the Gromov-Witten theory of the orbifold C/Z_r. Furthermore, we show that the quantum cohomology of C/Z_r is generically semisimple. Using recent ideas of Pandharipande-Pixton-Zvonkine, this semisimplicity may be useful for obtaining other tautological relations.

Iitaka's philosphy claims that whenever we have a theorem for complete varieties, we should have a counter-theorem for open varieties. In this talk, I will give an introduction on this philosophy with examples and evidence. Then I will explain the recent progress on the theory of rational curves on varieties under Iitaka's philosphy. This is a joint work with Qile Chen.

I will discuss joint work with Jack Huizenga and Matthew Woolf, where we describe the effective cone of the moduli spaces of semistable sheaves on the plane. The calculation is inspired by Bridgeland stability and hinges on the classification of stable vector bundles on the plane. The fractal nature of the classification and the remarkable number-theoretic properties of exceptional slopes play an essential role in the calculation.

We prove a new inversion of adjunction statement for rational and Du Bois singularities. Roughly speaking, this says that if we have a family over a smooth base with Du Bois special fiber and rational generic fiber, then the total space also has rational singularities. Furthermore, we even generalize this result to the context of rational and Du Bois pairs as defined by Kollár and Kovács. Imprecisely, a pair (X,D) is Du Bois if the failure of X to be Du Bois is equal to the failure of D to be Du Bois. In order to accomplish our inversion of adjunction result we need to prove, for pairs, many recent results on Du Bois singularities. I will describe some of these ideas. This is joint work with Sandor Kovács.

The fundamental work of Donaldson and Uhlenbeck-Yau proves the the smooth convergence of the Yang-Mills flow of stable integrable unitary connections on hermitian vector bundles over Kaehler manifolds. This was generalized by Bando and Siu to incorporate certain (singular) hermitian structures on reflexive sheaves. Bando-Siu also conjectured what happens when the initial sheaf is unstable; namely, that the limiting behavior should be controlled by the Harder-Narasimhan filtration of the sheaf. In this talk I will describe the solution to this question, which draws on the work of several authors.

Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting objects from simpler ones. I will illustrate how this works in representation theory and algebraic geometry via examples. I will explain a new structure theory, motivated by Hamiltonian reduction, for some categories (of D-modules) of interest to representation theorists, and, if there is time, indicate applications to the cohomology of (hyperkahler) manifolds. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.

The theory of stable log maps was developed recently for studying the degeneration of Gromov-Witten invariants. In this talk, I will introduce another important aspect of stable log maps as a useful tool for investigating A^1-curves on quasi-projective varieties, which are the analogue of rational curves on proper varieties. At least two interesting applications of A^1-curves will be introduced in this talk. For classical birational geometry, the A^1-curves can be used to produce very free rational curves on Fano complete intersections in projective spaces. On the arithmetic side, A^1-connectedness gives a general frame work for the existence of integral points over function field of curves. This is joint work with Yi Zhu.

I will discuss my joint with work Max Lieblich on Fourier-Mukai partners of K3 surfaces in positive characteristic. In particular, I will discuss the finiteness of the number of Fourier-Mukai partners of a given K3 surface, their realizations as moduli spaces of vector bundles, and various arithmetic applications.

Higher-rank Brill-Noether, at its most basic, seeks to answer the following question: how many global sections can a (semistable) vector bundle of given rank and degree have on general curve of genus g? When there exist vector bundles with a k-dimensional space of sections, one then asks how many such bundles there are. The classical rank-1 version was settled completely in the 1970's and 1980's, but even the rank-2 case is still wide open, with many partial results but no comprehensive conjecture. In the 1990's, Bertram, Feinberg and Mukai observed that the canonical determinant case has special behavior, with a higher expected dimension, and I will discuss recent work which studies the special determinant case more systematically. This work is in two parts: producing new expected dimensions on smooth curves by exploiting symmetries to produce lower bounds on dimensions, and developing the general theory of higher-rank limit linear series in order to use degeneration arguments to prove existence results. Some of this is joint work with Montserrat Teixidor i Bigas.

The Verlinde bundles over the moduli space M_g are obtained by considering relative moduli spaces of semistable bundles over smooth curves and associated theta divisors. Their ranks are given by the well-studied Verlinde numbers. I will discuss a formula for the total Chern character of the Verlinde bundles, as well as extensions over the moduli space of stable curves \overline M_g. This is based on joint work with Alina Marian, Rahul Pandharipande, Aaron Pixton and Dimitri Zvonkine.

Quasimaps provide compactifications, depending on a stability parameter, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well. The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

Let (X,H) be a ample polarization of a projective variety X. An H-polar cylinder in X is an open affine cylinder-like set whose complement is a support of an effective Q-divisor Q-rationally equivalent to H. This notion relates affine, birational and Kaehler geometry. we will show how to construct cylinders and to prove non-existence of them in smooth and mildly singular del Pezzo surfaces This is an extended answer of an question of Zaidenberg and Flenner in 2003. This is a joint work with I.Cheltsov and J.Park.

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In a modern language, it can be shown that the special groups are those of essential dimension zero. In 1958 Grothendieck classified special groups in the case where the base field k is algebraically closed. In this talk I will explain some recent progress towards the classification of special reductive groups over an arbitrary field. In particular, I will give the classification of special semisimple groups, special reductive groups of inner type and special quasisplit reductive groups over an arbitrary field k.

In a joint work with Giulia Saccà, we study the singularities of moduli spaces of rank 0 sheaves on K3 surface; we prove, in a number of cases, the formality of their Kuranishi family; and we relate their symplectic resolutions to the ones of Nakajima's quiver varieties coming from variations of GIT quotients.