Northwestern Algebraic Geometry Seminar, AY 2021--2022

Organizers: Jihao Liu, Yuchen Liu, Charlie Stibitz

The Northwestern Algebraic Geometry Seminar meets on Wednesdays from 3pm to 4pm Central Time. The in-person activities are held in Lunt 103. The online activities are held on Zoom. Please contact the organizers for Zoom information.

Schedule

  • Wed Jan. 5, Noah Olander (Columbia): TBA

    Abstract: TBA

  • Wed Jan. 26 (Zoom), Stefano Filipazzi (EPFL): TBA

    Abstract: TBA

  • Wed Feb. 2, Enrica Mazzon (Michigan): TBA

    Abstract: TBA

  • Wed Mar. 16, David Stapleton (Michigan): TBA

    Abstract: TBA

  • Past talks

  • Wed Sep. 29, Jihao Liu (Northwestern): Minimal model program for generalized lc pairs

    Abstract: The theory of generalized pairs was introduced by C. Birkar and D.-Q. Zhang in order to tackle the effective Iitaka fibration conjecture, and has proven to be a powerful tool in birational geometry. It has recently become apparent that the minimal model program for generalized pairs is closely related to the minimal model program for usual pairs and varieties. A folklore conjecture proposed by J. Han and Z. Li and recently re-emphasized by Birkar asks whether we can always run the minimal model program for generalized pairs with at worst lc singularities. In this talk, we will confirm this conjecture by proving the cone theorem, contraction theorem, and the existence of flips for generalized lc pairs. As an immediate consequence, we will complete the minimal model program for generalized lc pairs in dimension <=3 and the pseudo-effective case in dimension 4. This is joint work with C. D. Hacon.

  • Wed Oct. 6, Charlie Stibitz (Northwestern): Local Noether-Fano Type Inequalities

    Abstract: The Noether-Fano inequalities are a classical tool in birational geometry, which describe how a birational map of Fano varieties creates singular linear systems of divisors. In this talk, we provide a general form of the Noether-Fano inequalities which can in particular be applied to a local setting as well. These provide a numerical description of singular divisors obtained by examining two distinct prime blowups of a singularity.

  • Wed Oct. 20, Kevin Tucker (UIC): Splinter rings and Global +-regularity

    Abstract: A Noetherian ring is a splinter if it is a direct summand of every finite cover. Perhaps owing to their simple definition, basic questions about splinters are often devilishly difficult to answer. For example, Hochster's direct summand conjecture is the modest assertion that a regular ring of any characteristic is a splinter, and was finally settled by André in mixed characteristic more than three decades after Hochster's verification of the equal characteristic case using Frobenius techniques. In this talk, I will discuss some recent work on splinter rings in both positive and mixed characteristics. In particular, inspired by recent work of Bhatt on the Cohen-Macaulayness of the absolute integral closure, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This is based on joint works arXiv:2103.10525 with Rankeya Datta and arXiv:2012.15801 with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Joe Waldron, and Jakub Witaszek.

  • Wed Oct. 27 (Zoom), Linquan Ma (Purdue): Global F-regularity and global +-regularity

    Abstract: This is partially a continuation of Tucker's talk, on the other hand no prerequisites from his talk are required. We generalize the theory of globally F-regular pairs from positive to mixed characteristic, which we call global +-regularity, and introduce certain stable sections of adjoint line bundles. This is inspired by recent work of Bhatt on the Cohen-Macaulayness of the absolute integral closure. We will discuss applications of these results to birational geometry in mixed characteristic. Joint work with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.

  • Wed Nov. 3 (Zoom), Junpeng Jiao (Utah): DCC of Iitaka volume and boundedness of canonical models

    Abstract: The canonical models of varieties of general type are broadly understood thanks to the breakthroughs of Hacon-McKernan-Xu. In contrast, there are many open questions concerning canonical models of varieties of intermediate Kodaira dimensions. In this talk, we introduce the definition of Iitaka volume and use it to study the boundedness of canonical models under some natural conditions.

  • Wed Nov. 10 (Zoom), Dylan Spence (Indiana U Bloomington): Derived categories of singular curves

    Abstract: The derived category of a variety is a rich and interesting object which contains a significant amount of geometric information. However, analyzing the structure of this category can be difficult when the variety in question is singular. We discuss some recent work of the speaker which extends some results about the derived categories of smooth varieties to the singular setting, in particular, to singular projective curves.

  • Wed Nov. 17 (Zoom), Lena Ji (Michigan): The Noether-Lefschetz theorem in arbitrary characteristic

    Abstract: The classical Noether-Lefschetz theorem says that for a very general surface S of degree 4 in P^3 over the complex numbers, the restriction map from the divisor class group on P^3 to S is an isomorphism. In this talk, we will show a Noether-Lefschetz result for varieties over fields of arbitrary characteristic. The proof uses the relative Jacobian of a curve fibration, and it also works for singular varieties (for Weil divisors).

  • Mon Nov. 29, 4:10pm, Lunt 105 (Special time/location), Jakub Witaszek (Michigan): Classification of algebraic varieties in positive and mixed characteristic

    Abstract: In my talk I will describe recent developments in classifying algebraic varieties in positive and mixed characteristic. I will start by explaining the background for complex varieties and providing motivation for the study of their arithmetic analogues. The results which will be discussed in my talk are partially based on recent breakthroughs in arithmetic geometry and commutative algebra.

  • Wed Dec. 1, Kai Huang (MIT): K-stability of Log Fano Cone Singularities

    Abstract: In recent years, Fujita-Odaka, Li and Blum-Jonsson developed a valuative criterion for K-stability of Fano varieties. We generalize this criterion to the log Fano cone singularities. We also show the higher rank finite generation conjecture for log Fano cone singularities, which implies the Yau-Tian-Donaldson Conjecture for the Sasakian-Einstein metric.

  • COVID policy for in-person activities

    All attendees from outside Northwestern University will be required to have been fully vaccinated or have received a negative COVID test within 48 hours of the event start, as well as comply with all other University safety protocols that are in place at the time of the event. Participants unwilling or unable to abide by these requirements should not attend.