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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.