Algebra and Algebraic Geometry Seminar Spring 2020
Spring 2020 Schedule
|January 24||Xi Chen (Alberta)||Rational Curves on K3 Surfaces||Michael K|
|January 31||Janina Letz (Utah)||Local to global principles for generation time over commutative rings||Daniel and Michael B|
|February 7||Jonathan Montaño (New Mexico State)||Asymptotic behavior of invariants of symbolic powers||Daniel|
|February 21||Erika Ordog (Duke)||Minimal resolutions of monomial ideals||Daniel|
|March 13||Kevin Tucker (UIC)||TBD||Daniel|
|March 27||Patrick McFaddin (Fordham)||TBD||Michael B|
|March 30, 3:30pm (Monday, unusual date and time!)||Katrina Honigs (Oregon)||TBA||Andrei|
|April 10||Ruijie Yang (Stony Brook)||TBD||Michael K|
|April 24||Remy van Dobben de Bruyn (Princeton/IAS)||TBD||Botong|
|May 1||Lazarsfeld Distinguished Lectures|
Rational Curves on K3 Surfaces
It is conjectured that there are infinitely many rational curves on every projective K3 surface. A large part of this conjecture was proved by Jun Li and Christian Liedtke, based on the characteristic p reduction method proposed by Bogomolov-Hassett-Tschinkel. They proved that there are infinitely many rational curves on every projective K3 surface of odd Picard rank. Over complex numbers, there are a few remaining cases: K3 surfaces of Picard rank two excluding elliptic K3's and K3's with infinite automorphism groups and K3 surfaces with two particular Picard lattices of rank four. We have settled these leftover cases and also generalized the conjecture to the existence of curves of high genus. This is a joint work with Frank Gounelas and Christian Liedtke.
Local to global principles for generation time over commutative rings
Abstract: In the derived category of modules over a commutative noetherian ring a complex $G$ is said to generate a complex $X$ if the latter can be obtained from the former by taking finitely many summands and cones. The number of cones needed in this process is the generation time of $X$. In this talk I will present some local to global type results for computing this invariant, and also discuss some applications of these results.
Asymptotic behavior of invariants of symbolic powers
Abstract: The symbolic powers of an ideal is a filtration that encodes important algebraic and geometric information of the ideal and the variety it defines. Despite the importance and great results about symbolic powers, their complete structure is far from being understood. For example, we do not completely understand yet the behavior of the number of generators, regularities, and depths of these ideals. In this talk I will report on resent results in this direction in joint works with Hailong Dao and Luis Núñez-Betancourt.
Minimal resolutions of monomial ideals
Abstract: The problem of finding minimal free resolutions of monomial ideals in polynomial rings has been central to commutative algebra ever since Kaplansky raised the problem in the 1960s and his student, Diana Taylor, produced the first general construction in 1966. The ultimate goal along these lines is a construction of free resolutions that is universal -- that is, valid for arbitrary monomial ideals -- canonical, combinatorial, and minimal. This talk describes a solution to the problem valid in characteristic 0 and almost all positive characteristics.
An Analog of PLT Singularities in Mixed Characteristic
Abstract: In algebraic geometry, singularities are often understood using hyperplane sections. For example, if the singularities of a given hyperplane section are mild, one can ask whether the same holds for the ambient variety. Such inversion of adjunction questions have a fairly satisfactory answer in both equal characteristic zero and p>0, and in this talk we aim to address this problem in mixed characteristic. Using the framework of perfectoid big Cohen-Macaulay algebras, we define a class of singularities satisfying adjunction and inversion of adjunction properties analogous to those for PLT singularities in complex algebraic geometry (or purely F-regular singularities in characteristic p>0). As an application, we obtain a new form of the Briancon-Skoda theorem in mixed characteristic.