Algebraic Geometry Seminar

A few years ago, Chi Li introduced the notion of local volume of Kawamata log terminal (klt) singularities as the minimum normalized volume of valuation. This invariant carries lots of interesting geometric information of the singularity, for instance: it characterizes smooth points; it detects orbifold order of quotient singularities; it is bounded from above by the minimal log discrepancy. In this talk, I will discuss the conjecture that local volumes of klt singularities in a fixed dimension with finite coefficient set has only accumulation point zero. We confirm this conjecture when ambient singularities are bounded. This is a joint work in progress with Jingjun Han and Lu Qi.

We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.

The McKay correspondence takes many guises but at its core connects the geometry of minimal resolutions for quotient singularities C^n / G to the representation theory of the group G. When G is an abelian subgroup of SL(3), Craw-Ishii showed that every minimal resolution can be realised as a moduli space of stable quiver representations naturally associated to G, although the chamber structure for the stability parameter and associated wall-crossing behaviour is relatively poorly understood. I will describe recent work giving explicit representation-theoretic descriptions of the walls and wall-crossing behaviour for the chamber corresponding to a particular minimal resolution called the G-Hilbert scheme. Time permitting, I will also discuss ongoing work with Yukari Ito (IPMU) and Tom Ducat (Imperial) to better understand the geometry, chambers, and corresponding representation theory for other minimal resolutions.

Del Pezzo fibrations appear naturally as one of the possible outcomes of the Minimal Model Program for threefolds. In this talk, I will discuss some 'pathologies' that can appear in characteristic p and how it is possible to bound the bad behaviour of such fibrations depending on the prime p. This is joint work with H. Tanaka.

By Raynaud's result, Akizuki-Nakano vanishing theorem holds on a smooth globally F-split variety if the characteristic is larger than or equal to the dimension. However, very little is known for singular varieties. In this talk, we first show a weak form of the Akizuki-Nakano vanishing theorem on (possibly singular) globally F-split 3-folds. As application, we obtain results on deformations for globally F-split Fano 3-folds. We also apply it to prove the Kodaira vanishing for thickenings of locally complete intersection globally F-regular 3-folds, which is a positive characteristic analogue of a result by Bhatt-Blickle-Lyubeznik-Singh-Zhang. This talk is based on joint work with Shunsuke Takagi.

I will discuss a class of singularities defined using the Big Cohen-Macaulay algebras coming out of work of Andre, Bhatt and Gabber. These are a mixed characteristic analog of KLT and strongly F-regularity singularities. I will focus on some new work which lets us show large classes of singularities are BCM regular.

This is a joint work with Bumsig Kim. We define the notion of Atiyah class for global matrix factorization and use it to calculate the categorical Chern character. The key role is played by the analog in this context of the Lie algebra structure on the shifted tangent bundle defined and studied by Kapranov and Markarian.