## Conference

### Schedule

#### Monday

- 9:00 -9:45 Vakil
- 9:45-10:15 Coffee
- 10:15-11:00 Berkesch
- 11:00-1:00 Lunch
- 1:00-1:45 Caviglia
- 2:00-2:45 Peeva
- 2:45-3:15 Coffee
- 3:15-4:00 Hochster

#### Tuesday

- 9:00-9:45 Seceleanu
- 9:45-10:15 Coffee
- 10:15-11:00 McCullough
- 11:00-1:00 Lunch
- 1:00-1:45 Mapes
- 2:00-2:45 Galligo
- 2:45-3:15 Coffee
- 3:15-4:00 Schreyer
- 6pm Conference Dinner

All talks will be in room 150 of the Goldman School of Public Policy (GSPP), which is near Evans Hall.

### Abstracts

- Christine Berkesch Zamaere - Virtual resolutions for products of projective spaces

Given a module M over the Cox ring of a smooth toric variety, one can consider free complexes that are acyclic modulo irrelevant homology, which we call virtual resolutions for M. These complexes have many advantages over minimal free resolutions over smooth toric varieties other than projective spaces. We develop this in detail for products of projective spaces. This is joint work with Daniel Erman and Gregory G. Smith.

- Giulio Caviglia - Extremal free resolutions over monomial complete intersections.

Let S be a polynomial ring over a field k of characteristic 0, and let R be a complete intersection in S defined by powers of the variables. We prove that, in the Hilbert scheme parametrizing the closed subschemes of Proj R with a fixed Hilbert polynomial p,there exists a point whose saturated ideal I achieves the largest possible Betti numbers in the finite free resolution of R/I over S and in the infinite free resolution of R/I over R. In the case of a regular ring, the ideal I also maximizes the infinite free resolution of k over R/I. This is a joint work with Alessio Sammartano.

- Andre Galligo - Deformations of curve germs with and without embedded components.

Recent articles (by J. Bobadilla, J. Snoussi, M. Spivatovski and by G-M. Greuel) provided interesting generalizations of some results of my old joint paper with J. Brian_c{c}on and J-M. Granger, which classified the natural notions of equisingularity for deformations of reduced complex curve germs. They studied reduced surface germs S, deformations of curve germs which are only generically reduced. One of the consequences is that such an S is not necessarily Cohen-Macaulay. I will first present the topological and algebraic aspects of the subject. Then I will report on experiments for an attempt to recover some results using Macaulayfication (CM linkage) of the surface S.

- Mel Hochster - Small subalgebras and Stillman bounds

The talk will discuss joint work of the speaker and Tigran Ananyan proving Stillman’s conjecture for all degrees and characteristics, developing specific bounds for the functions involved in small degree by somewhat different methods, as well as some related developments.

- Sonja Mapes - Computing projective dimension of monomial ideals

Given a square-free monomial ideal I in a polynomial ring R over a field k, we would like to know the projective dimension of I. We recall the definition of the lcm-lattice of a monomial ideal introduced by Gasharov, Peeva and Welker, and the definition of the dual hypergraph of a square-free monomial ideal introduced by Kimura, Terai and Yoshida. Our work focuses on the relationship between the lcm-lattice and the dual hypergraph of a given square-free monomial ideal. We use the properties of lcm-lattice to find whether two different dual hypergraphs have the same projective dimension, and thus are able to extend some of the results by Lin and Mantero which compute the projective dimensions for ideals with certain hypergraphs. This is joint work with Kuei-Nuan Lin.

- Jason McCullough - The Projective Dimension of 3 Cubics

Stillman’s Question, recently answered in the affirmative by Ananyan-Hochster, asks whether there exists a bound on the projective dimension of homogeneous ideals in polynomial rings over a field depending only on the degrees of the generators. While the Ananyan-Hochster result shows there is such a bound in all cases, what the optimal answer is remains unknown in almost all cases. Previously Engheta had shown that if I = (f,g,h) is an ideal in a polynomial ring S generated by 3 cubic forms, then the projective dimension of S/I is at most 36, while the largest known example had projective dimension 5. I will sketch the proof that 5 is indeed the maximum projective dimension of S/I, where I is generated by 3 cubics. This is joint work with Paolo Mantero.

*Irena Peeva - Codimension two complete intersections

We will discuss the structure of minimal free resolutions and of Ulrich modules over a codimension two complete intersection ring. This is joint work with David Eisenbud.

- Frank-Olaf Schreyer - Balancing in relative canonical resolutions and a unirational moduli space of K3 surfaces

I report on work of two of my students: Christian Bopp and Michael Hoff. Given a curve C of genus g together with a rational function f:C -> P^1 of degree d the canonical model lies on a rational normal scroll X, and the resolution of O_C as an O_X module is build with certain vector bundle N_i on P^1. It is interesting to ask whether the splitting type of the N_i is balanced for a general pair (C,f), since then jump loci lead to interesting subspaces of the Hurwitz scheme H_{g,d}. By experiment Bopp and Hoff discovered that the second syzygy bundle N_2 is not balanced for (g,d)=(9,6) for finite fields. In the talk I will explain how their proof in characteristic zero builds upon a moduli space of certain lattice polarized K3 surfaces.

- Alexandra Seceleanu - Polynomial growth for Betti numbers

It is well known that the asymptotic patterns of the Betti sequences of the finitely generated modules over a local ring R reflect the structure of R. For instance, these sequences are eventually zero if and only if R is regular and they are eventually constant if and only if R is a hypersurface. We consider the problem of characterizing the rings R such that every R-module has Betti numbers eventually given by a single polynomial. We give necessary and sufficient conditions for R to have this property. In some important cases, for example when R is homogeneous, these conditions coincide and therefore characterize R.

This is joint work with Lucho Avramov and Yang Zheng.

- Ravi Vakil - Limits of Hurwitz spaces (in various senses)

I will discuss three projects involving Hurwitz spaces, all with the hope of some sort of structure stabilizing as the genus or gonality becomes large. First (with Matchett Wood, in progress): the class of curves of gonality 3 and 4 (and hopefully 5) in the Grothendieck ring tends to a limit as the genus gets large, analgous to Bhargava’s counts of number fields, also in terms of zeta-values. Second (with Patel): the Chow ring of trigonal curves with simple branching is trivial. Third (very much in progress, with Deopurkar and Patel): the class of genus g degree d simply branched covers of the projective line stabilizes as d gets large; the punchline is that one might hope that the limit is a particularly nice answer.