• Speaker: Laure Flapan, MSU
• Title: Fano manifolds associated to hyperkähler manifolds
• Date: 02/03/2021
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Laure Flapan (flapanla@msu.edu)

Many of the known examples of hyperkähler manifolds arise from geometric constructions that begin with a Fano manifold whose cohomology looks like that of a K3 surface. In this talk, I will focus on a program whose goal is to reverse this process, namely to begin with a hyperkähler manifold and from it produce geometrically a Fano manifold. This is joint work in progress with K. O’Grady, E. Macrì, and G. Saccà. Passcode: MSUALG

• Speaker: François Greer, IAS
• Title: A tale of two Severi curves
• Date: 02/17/2021
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Laure Flapan (flapanla@msu.edu)

Let $(S,L)$ be a general polarized K3 surface with $c_1(L)^2=2g-2$. A general member of the linear system $|L|\simeq \mathbb P^g$ is a smooth curve of genus $g$. For $0\leq h\leq g$, define the Severi variety $V_h(S,L)\subset |L|$ to be the locus of curves with geometric genus $\leq h$. As expected, $V_h(S,L)$ has dimension $h$. We consider the case $h=1$, where the Severi variety is a (singular) curve. Our first result is that the geometric genus of $V_1(S,L)$ goes to infinity with $g$; we give a lower bound $\sim e^{c\sqrt{g}}$. Next we consider the analogous question for Severi curves of a rational elliptic surface, and give a polynomial upper bound instead. Modular forms play a central role in both arguments. Passcode: MSUALG

• Speaker: Rankeya Datta, University of Illinois at Chicago
• Title: Openness of splinter loci in prime characteristic.
• Date: 03/03/2021
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Joe Waldron (waldro51@msu.edu)

A splinter is a notion of singularity that has seen numerous applications of late, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and vanishing theorems. However, many basic questions about splinters remain elusive. One such problem is whether the splinter condition spreads from a point to an open neighborhood of a noetherian scheme. In this talk, we will address this question in prime characteristic and show that a locally noetherian scheme whose associated absolute Frobenius is finite map has an open splinter locus. In particular, all varieties over perfect fields of positive characteristic have open splinter loci. If time permits, we will show how our methods also give openness of splinter loci for a large class of schemes that do not necessarily have finite Frobenius. This talk is based on joint work in progress with Kevin Tucker.

• Speaker: Daniel Bragg, University of California, Berkeley
• Title: TBA
• Date: 03/17/2021
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting)
• Contact: Rajesh S Kulkarni (kulkar23@msu.edu)

No abstract available.

• Speaker: Oscar Rivero, University of Warwick
• Title: TBA
• Date: 03/31/2021
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting)
• Contact: Preston Wake (wakepres@msu.edu)

No abstract available.

• Speaker: Yihang Zhu, University of Maryland
• Title: TBA
• Date: 04/14/2021
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting)
• Contact: Igor Rapinchuk (rapinchu@msu.edu)

No abstract available.