Algebra and Algebraic Geometry Seminar Spring 2019
The seminar meets on Fridays at 2:25 pm in room B235.
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Spring 2019 Schedule
|January 25||Daniel Smolkin (Utah)||Symbolic Powers in Rings of Positive Characteristic||Daniel|
|February 1||Juliette Bruce||Asymptotic Syzgies for Products of Projective Spaces||Local|
|February 8||Isabel Vogt (MIT)||Low degree points on curves||Wanlin and Juliette|
|February 15||Pavlo Pylyavskyy (U. Minn)||TBD||Paul Terwilliger|
|February 22||Michael Brown||Chern-Weil theory for matrix factorizations||Local|
|March 8||Jay Kopper (UIC)||TBD||Daniel|
|March 22||No Meeting||Spring Break||TBD|
|March 29||Chris Eur (UC Berkeley)||TBD||Daniel|
|April 19||Eloísa Grifo (Michigan)||TBD||TBD|
Symbolic Powers in Rings of Positive Characteristic
The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!
Title: Asymptotic Syzygies for Products of Projective Spaces
I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.
Title: Low degree points on curves
In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.