## BAGELS TALKS - Spring 2023

During Spring 2023 semester we meet on Thursdays at 3:00 - 4:00 on JWB 208. If you are around Salt Lake City and want to participate, contact me!

Date

Speaker

Title/Abstract

January 19th Organizational Meeting

January 26th Lingyao Xie Minimal Model Program for Generalized PairsTitle:Generalized pairs were introduced by Birkar and Zhang in [BZ16] in their study of effective Iitaka fibrations, and later became a central topic in modern day birational geometry. For exmaple, the theory of generalized pairs is used to prove the Borisov-Alexeev-Borisov conjecture [Bir19,Bir21]. In this talk I will explain why we care about generalized pairs by showing that they naturally appear in the classification of algebraic varieties. Then I will discuss several recent fundamental results about generalized pairs. e.g Cone theorem ([Hancon-Liu21]), Contraction theorem ([Xie22]), Existence of flips ([Liu-Xie22]). If time permits, I will give the sketch of the proofs, and say something about Koll\'ar's gluing theory, which is a key ingredient in our proofs. As an interesting corollary, we show that glc singularities are Du Bois.Abstract:

February 2nd Qingyuan Xue Generalized cone and rationality theoremTitle:In this talk, I will introduce the generalized cone theorem (by Choi and Gongyo) which describes the cone of curves that are movable in codimension l. As for its proof, I will focus on a generalized rationality theorem, which is the key step.Abstract:

February 23rd Seungsu Lee Global F-regular varieties are log Fano.Title:In this talk, we will discuss globally F-regular/F-split pairs and will show that global F-regular varieties are log Fano, which is proved by Schwede and Smith (https://arxiv.org/pdf/0905.0404.pdf). Furthermore, we will discuss an open question in this direction.Abstract:

April 13th Yu-Ting Huang Schubert Calculus on Grassmann BundleTitle:Schubert calculus is a technique developed in the nineteenth century that plays an important role in enumerative geometry and algebraic combinatorics. Hilbert's 15th problem put Schubert's calculus on a rigorous foundation. In this talk, I will introduce the construction of Schubert calculus on Grasmmanian and its generalization on Grassmann bundles. Then we will see their application in enumerative geometry.Abstract:

April 20th Yi-Heng Tsai Multiplier Ideals and V-filtrationTitle:Given an effective divisor on a smooth complex algebraic variety X, we have a decreasing filtration on O_X associated to the multiplier ideals of cD, where c is a rational number. In this talk, following the approaches in "multiplier ideals, V-filtration and spectrum", written by N. Budur and M. Saito, I will establish the equivalence between the filtration given by the multiplier ideal and the V-filtration on X. One direct application is that the equivalence gives another proof of a theorem L. Ein, R Lazarsfeld, K.E. Smith and D. Varolin that the jumping numbers of multiplier ideals in (0,1] are roots of the Bernstein-Sato polynomial up to sign. To achieve the goal, I will briefly review the definitions of multiplier ideals and D-modules, and then compute the V-filtration corresponding to a normal crossing divisor.Abstract:

## Past seminars