Algebra Seminar

Organizers: Justin Lyle and Paolo Mantero
Time: 3:30pm virtual (Email the organizers for Zoom access).

Feb. 4
C. Eric Overton-Walker
Derived algebraic geometry and jet schemes
To any scheme X one can define a jet scheme J^m X whose points should be thought of as points in X along with tangent directions up to order m. This perspective lends itself well to the setting of derived algebraic geometry, where deformation theory is more tractable. In this talk, we'll discuss the derived setting via some well-known constructions, then discuss how one can upgrade the classical jet construction to the derived realm. Time permitting, we'll see that smoothness forces derived jet spaces to be homotopically discrete, suggesting the construction is a tool that can measure degree of singularity. Joint with Lance Edward Miller.
Mar. 4
Alexander Duncan
Sharpening the Gonality Theorem using N_p Properties
It’s well known that one can read-off the gonality of a projective curve embedded by a sufficiently high degree line bundle by finding the last nonzero entry of the middle row of the corresponding Betti diagram. This is the content of the gonality theorem. However, there are only conjectures for sharp sufficient conditions on degree. Given a globally generated line bundle on a projective variety, one may surject generators onto this line bundle and this defines a kernel bundle. Mild cohomological vanishing conditions of kernel bundles become equivalent to certain N_p properties (these are geometric properties that are stronger than k-normality for all k). In this talk, we will show how one can utilize N_p properties of line bundles on the veronese surface to find examples of curves which sharpen the degree bounds of the gonality theorem.
Mar. 18
Matt Mastroeni
(Iowa State)
Chow rings of matroids are Koszul
The Chow ring of an algebraic variety is an algebro-geometric analog of the cohomology ring of a smooth manifold that encodes important information about the intersections between its subvarieties. Feichtner and Yuzvinsky computed a presentation for the Chow ring of a smooth toric variety associated to a matroid (and some other data) which is now called the Chow ring of the matroid. These rings have garnered significant attention in recent years thanks to their role in establishing long-standing conjectures on the combinatorics of matroids, including the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. From a commutative algebra standpoint, Chow rings of matroids are very nice graded Artinian Gorenstein rings defined by quadratic relations, and so, a natural conjecture posed by Dotsenko is that the Chow ring of a matroid is always Koszul. In this talk, we will discuss how the combinatorics of a matroid influences algebraic properties of its Chow ring, culminating in recent joint work with Jason McCullough giving an affirmative answer to Dotsenko’s conjecture.