Algebraic Geometry Seminar

Since Bernstein—Gelfand—Gelfand’s analysis of the ring of differential operators on the cone over an elliptic curve, geometric reasoning has provided one of the few known methods for the difficult algebraic problem of describing the differential operators on a singular ring in characteristic 0. Attempts to study similar phenomena in higher dimension lead naturally to the study of positivity (bigness) of the tangent bundle of Fano varieties, a question which is of independent interest and which parallels well-known theorems and conjectures regarding the ampleness and nefness of the tangent bundle. In this talk, we will review recent progress on this problem, and (more significantly) the questions that remain unknown. We’ll also point out the consequence this study has for several phenomena regarding reduction to positive characteristic, including questions about associated primes of local cohomology modules in mixed characteristic and the question of reduction of log canonical singularities mod \(p\).

Hyperelliptic Hodge integrals are a class of intersection numbers on the moduli space of hyperelliptic curves. Several mathematicians (Graber-Pandharipande, Cavalieri, Wise, Johnson-Pandharipande-Tseng) have investigated these intersection numbers in the past, and their work has made it apparent that hyperelliptic Hodge integrals display a remarkable amount of combinatorial structure and symmetry. However, these intersection numbers are difficult to compute in general. In this talk, I will explain some results about closed form expressions for hyperelliptic Hodge integrals, culminating in a structure theorem (polynomiality) for a very general class of Hodge integrals.

For a very general polarized K3 surface over the rational numbers, it is a consequence of the Tate conjecture that the Picard rank jumps upon reduction modulo any prime. This jumping in the Picard rank is countered by a drop in the size of the Brauer group. In this talk, I will report on joint work with Brendan Hassett and Anthony Várilly-Alvarado, in which we consider the following: Given a non-trivial Brauer class on a very general polarized K3 surface over Q, how often does this class become trivial upon reduction modulo various primes? This has implications for the rationality of reductions of cubic fourfolds as well as reductions of twisted derived equivalent K3 surfaces

The minimal log discrepancy introduced by Shokurov is a basic invariant in birational geometry. Shokurov conjectured that the set of threefold minimal log discrepancies should satisfy the ascending chain condition. This conjecture has a close relation with the termination of flips in the minimal model program. In this talk, I will report on our recent progress towards Shokurov’s conjecture for threefolds. Some applications will also be discussed. This is a joint work in progress with Jihao Liu and Yujie Luo.

Inspired by Deligne’s use of the simplicial theory of hypercoverings in defining mixed Hodge structures we replace the indexing category ∆ by the symmetric simplicial category ∆S and study (a class of) ∆S-hypercoverings, which we call spaces admitting symmetric (semi)simplicial filtration. For ∆S-hypercoverings we construct a spectral sequence, somewhat like the Cˇech-to-derived category spectral sequence. The advantage of working on ∆S is that all of the combinatorial complexities that come with working on ∆ are bypassed, giving simpler, unified proof of known results like the computation of (in some cases, stable) singular cohomology (with rational coefficients) and étale cohomology (with Q_l coefficients) of the moduli space of degree \(n\) maps \(C\) to a projective space, \(C\) a smooth projective curve of genus \(g\), of unordered configuration spaces, that of the moduli space of smooth sections of a fixed \(g_d^r\) that is \(m\)-very ample etc.

The Serre functor of a triangulated category is one of its most important invariants, playing the role of the dualizing complex of a variety in noncommutative algebraic geometry. I will explain how to describe the Serre functors of many semiorthogonal components of varieties in terms of spherical twists. In the case of Kuznetsov components of Fano complete intersections, this leads to a proof of a conjecture of Katzarkov and Kontsevich on the dimensions of such categories, and implies the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide. This is joint work with Alexander Kuznetsov.