About 

This is a one day algebraic geometry event, happening on Saturday, April 5, 2025 at the University of Michigan. 

Schedule 

All talks will be held in 1360 East Hall. Refreshments will be available in the atrium. 

9:30-10:00 Refreshments 

10:00-10:50 Bochao Kong

11:10-12:00 Dmitry Zakharov

12:00-2:00 Lunch break

2:00-3:00 Burt Totaro

3:00-3:30 Refreshments 

3:30-4:00 Saket Shah 

4:30-5:00 Shikha Bhutani

6:00- Conference dinner 

Speakers and Abstracts

• Shikha Bhutani (Michigan State University) 

On Kawamata-Viehweg vanishing for surfaces of del-Pezzo type over imperfect fields

We prove the Kawamata-Viehweg vanishing for surfaces of del Pezzo type over imperfect fields of characteristic p > 5. Consequently, it follows that klt singularities on excellent threefold are rational.

• Bochao Kong (Michigan State University)

On Certain Degenerations of Elliptic Surfaces

The degeneration of surfaces is an important topic related to the compactification of moduli spaces, the study of singularities, period maps, and more. The most fruitful case is that of K-trivial degenerations, studied by Kulikov, Persson, and Pinkham, where a classification result with concrete dual complex homotopy types is available. We construct a particular type of degenerations of elliptic surfaces with interesting dual complex homotopy types. This should be seen as an analogue of Type III degenerations in the K-trivial case.

• Saket Shah (University of Michigan)

Generalized Galkin-Shinder-Voisin flips

Let X be a smooth cubic hypersurface. In 2014, Galkin and Shinder discovered a beautiful relation in the Grothendieck ring K_0(Var) relating the Hilbert scheme of 2 points in X and the Fano variety of lines in X, arising from a birational map later studied further by Voisin. I will present a generalization of this birational correspondence in two directions, and mention connections to semiorthogonal decompositions of derived categories. 

• Burt Totaro (UCLA) 

Endomorphisms of varieties 

A natural class of dynamical systems is obtained by iterating polynomial maps, which can be viewed as maps from projective space to itself. One can ask which other projective varieties admit endomorphisms of degree greater than 1. This seems to be an extremely restrictive property, with all known examples coming from toric varieties (such as projective space) or abelian varieties. We describe what is known in this direction, with the new ingredient being the "Bott vanishing" property. Joint work with Tatsuro Kawakami.

• Dmitry Zakharov (Central Michigan University)

The tropical trigonal construction

There are two standard ways to associate a principally polarized abelian variety (ppav) to a smooth algebraic curve X of genus g. The Jacobian variety Jac(X) is a ppav of dimension g. An etale double cover X’->X determines the Prym variety Prym(X’/X), which is a ppav of dimension g-1. These two objects are related by Recillas’ trigonal construction: given an etale double cover X’->X of a trigonal curve X, we can construct a tetragonal curve Y such that Prym(X’/X) is isomorphic to Jac(Y).

I will talk about a tropical version of the trigonal construction, where algebraic curves are replaced by metric graphs and ppavs by real tori with integral structure. Given a double cover X’->X of a trigonal graph X, we obtain a tetragonal graph Y such that the tropical Prym variety Prym(X’/X) and the tropical Jacobian Jac(Y) are isomorphic.

This construction has two applications. First, we can use it to compute the second moment of the tropical Prym variety for g up to 4, and conjecturally for all g, which has arithmetic applications. Second, the tropical trigonal construction provides an explicit resolution of the Prym—Torelli map in genus 4.

Registration 

If you would like to attend, please fill out the registration form, where you can indicate whether you would like to attend the conference dinner or be considered for travel funding. 

The registration deadline is March 15, 2025 for those who wish to attend the conference dinner, and March 28, 2025 otherwise. 

Contact

Please send any questions to arper@umich.edu.