Algebraic Geometry Seminar

Spring 2019 — 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
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 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.
January 21st
January 28th
Frebruary 4th
February 11th Ben Wormleighton
UC Berkeley
February 18th
February 25th Kenta Sato
Akizuki-Nakano vanishing on globally F-split 3-folds and its application
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.
March 3rd
March 17th Alexander Polishchuk
University of Oregon
March 24th
March 31st
April 7th
April 14th
April 21st

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