Algebraic Geometry Seminar
Spring 2020 — Tuesdays 3:30  4:30 PM, location LCB 222
Date  Speaker  Title — click for abstract (if available) 
January 8th LCB 225 (Wednesday, special time & place) 
Yuchen Liu Yale 
Discreteness of local volumes
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.

January 14th 
Carl Lian Columbia University 
Enumerating pencils with moving ramification on curves
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 basepoints are allowed. We then deduce, using an inclusionexclusion 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 FarkasMoschettiNaranjoPirola.

January 21st 


January 28th 


Frebruary 4th 


February 11th (special time and place: 1:302:30 in JWB 208) 
Ben Wormleighton UC Berkeley 
McKay correspondence and walls for GHilb
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), CrawIshii 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 wallcrossing behaviour is relatively poorly understood. I will describe recent work giving explicit representationtheoretic descriptions of the walls and wallcrossing behaviour for the chamber corresponding to a particular minimal resolution called the GHilbert 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.

February 18th 
Fabio Bernasconi University of Utah 
On del Pezzo fibrations in positive characteristic
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.

February 25th 
Kenta Sato RIKEN 
AkizukiNakano vanishing on globally Fsplit 3folds and its application
By Raynaud's result, AkizukiNakano vanishing theorem holds on a smooth globally Fsplit 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 AkizukiNakano vanishing theorem on (possibly singular) globally Fsplit 3folds.
As application, we obtain results on deformations for globally Fsplit Fano 3folds.
We also apply it to prove the Kodaira vanishing for thickenings of locally complete intersection globally Fregular 3folds, which is a positive characteristic analogue of a result by BhattBlickleLyubeznikSinghZhang.
This talk is based on joint work with Shunsuke Takagi.

March 3rd 
Karl Schwede Univeristy of Utah 
Perfectoid BCMRegular Singularities
I will discuss a class of singularities defined using the Big CohenMacaulay algebras coming out of work of Andre, Bhatt and Gabber. These are a mixed characteristic analog of KLT and strongly Fregularity singularities. I will focus on some new work which lets us show large classes of singularities are BCM regular.

March 17th 
Alexander Polishchuk University of Oregon 
Atiyah classes for global matrix factorizations
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.

March 24th 
Stefano Filipazzi UCLA 

March 31st 
Devlin Mallory University of Michigan 

April 7th 
Sean Howe University of Utah 

April 14th 
Yoshinori Gongyo University of Tokyo 

April 21st 
Alicia Lamarche University of South Carolina 

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