Moduli Workshop: Titles and Abstracts

Volumes of surface pairs of general type

by Valery Alexeev (Georgia)
Thursday, 8 December 2016 at 16:00 in Room 230 at the Fields Institute

Let X be a smooth projective surface and B a divisor with normal crossings such that L = KX + B is big. What is the smallest possible volume of L? The answer to this and related questions has many applications to compact moduli of surfaces of general type and to automorphisms of surfaces. In a joint work with Wenfei Liu, we improve several existing records. The new examples are based on a certain combinatorial problem which I will explain.

Enumerating finite Quot Schemes on Curves and Surfaces

by Aaron Bertram (Utah)
Friday, 9 December 2016 at 15:00 in Room 230 at the Fields Institute

Let X be a smooth projective curve of postive genus or else a del Pezzo surface and let Q(V,c) be the Grothendieck Quot scheme associated to a vector bundle V and a Chern class c (of the quotient) on X. For some choices of V and c this is a finite, reduced scheme that is therefore something we might enumerate. When X is a curve this was done by Marian and Oprea, and the answer is given by the Verlinde formula. When X is a del Pezzo surface, there seems to be a similar connection with Goettsche-type formulas that begs a deeper understanding. This is joint work with Thomas Goller and Drew Johnson.

The derived category of moduli spaces of pointed stable rational curves

by Ana-Maria Castravet (Northeastern)
Tuesday, 6 December 2016 at 09:30 in Room 230 at the Fields Institute

I will report on joint work with Jenia Tevelev on Orlov’s question on exceptional collections on moduli of pointed stable rational curves.

Hyperelliptic loci and Weierstrass points

by Renzo Cavalieri (Colorado State)
Monday, 5 December 2016 at 9:30 in Room 230 at the Fields Institute

The closure of the hyperelliptic locus is a cycle whose points parameterize curves admitting a double cover of the projective line. Although it’s a classical locus in the moduli space of curves, relatively little is still known about its class in the tautological ring. In this talk we will introduce and explore the notion of graph formula for a tautological class( a là Pixton). We will then discuss recent ongoing work with Tarasca, and (time permitting) Clader and Ross on the class of the hyperelliptic locus and on the class of hyperelliptic curves with marked Weierstrass points.

Topology of the tropical moduli spaces of curves

by Melody Chan (Brown)
Monday, 5 December 2016 at 16:00 in Room 230 at the Fields Institute

The moduli space of n-marked, genus g tropical curves is a cell complex that was identified in work of Abramovich-Caporaso-Payne with the boundary complex of the complex moduli space Mg,n. I will give results on the topology of tropical M1,n, joint with Galatius and Payne, and on tropical M2,n, obtaining as corollaries new calculations on the top-weight cohomology of the complex moduli spaces M1,n and M2,n.

Double ramification cycles and tautological relations

by Emily Clader (SFSU)
Wednesday, 7 December 2016 at 11:00 in Room 230 at the Fields Institute

Tautological relations are certain equations in the Chow ring of the moduli space of curves. I will discuss a family of tautological relations, first conjectured by A. Pixton, that arises by studying moduli spaces of ramified covers of the projective line. These relations can be used to recover a number of well-known facts about the moduli space of curves, as well as to generate very special equations known as topological recursion relations. This is joint (and partially ongoing) work with various subsets of S. Grushevsky, F. Janda, X. Liu, X. Wang, and D. Zakharov.

Invariant theory of Artinian Gorenstein algebras

by Maksym Fedorchuk (Boston College)
Thursday, 8 December 2016 at 14:30 in Room 230 at the Fields Institute

I will discuss the interplay between hypersurface singularities, their Milnor algebras, and classical invariant theory of homogeneous forms. In particular, I will prove that a contravariant that associates to a smooth homogeneous form the Macaulay inverse system of its Milnor algebra preserves GIT stability. I will discuss some applications of this result, for example to the direct sum decomposability of polynomials, and many related open problems.

Recent developments in tropical scheme theory

by Noah Giansiracusa (Swarthmore)
Friday, 9 December 2016 at 13:30 in Room 230 at the Fields Institute

Tropical scheme theory is a method of describing tropical varieties with equations, in order to incorporate more foundations and constructions from modern algebraic geometry into the subject. I’ll give an overview of this topic, emphasizing recent connections to combinatorics and hints of moduli theory that arise.

Weak Brill-Noether for rational surfaces

by Jack Huizenga (Penn State)
Tuesday, 6 December 2016 at 16:00 in Room 230 at the Fields Institute

A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Goettsche and Hirschowitz prove that on the projective plane every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also use combinatorial methods to prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef. This is joint work with Izzet Coskun.

Linear Systems on General Curves of Fixed Gonality

by Dave Jensen (Kentucky)
Monday, 5 December 2016 at 14:30 in Room 230 at the Fields Institute

The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

Conformal block vector bundles on moduli space of stable curves with marked points

by Anna Kazanova (Tennessee)
Tuesday, 6 December 2016 at 14:30 in Room 230 at the Fields Institute

Conformal block vector bundles are vector bundles on the moduli space of stable curves with marked points defined using certain Lie theoretic data. Over smooth curves, these vector bundles can be identified with generalized theta functions. In this talk we discuss extension of this identification over the stable curves.

Local and global equations of the Hilbert scheme

by Paolo Lella (Trento)
Tuesday, 6 December 2016 at 11:00 in Room 230 at the Fields Institute

The Hilbert scheme parametrizing subschemes and flat families of subschemes of a given projective space with fixed Hilbert polynomial is a central object in the study of algebraic varieties. In general, it is very difficult to determine explicitly equations defining a Hilbert scheme, since it is constructed as subschemes of a Grassmannian and then of a projective space of very large dimension through the Plücker embedding. In my talk, I will present a new type of flat families, called marked families, that are well suited to compute local and global equations of the Hilbert scheme.

Marked families are a generalization of another type of families called Groebner strata. Consider a polynomial ring with n+1 variable and coefficients in an algebraically closed field of characteristic zero and fix a term ordering sigma. For every monomial ideal I, we define the Groebner stratum of I and sigma as the set of homogeneous ideals having initial ideal I w.r.t. sigma. The marked family of I consists of homogeneous ideals whose quotient algebra is generated as a vector space by the set of monomials not contained in I and, in general, it is larger than every Groebner stratum of I. Groebner strata have a natural structure of affine homogeneous varieties. Their equations can be efficiently computed by using Buchberger’s algorithm to impose that syzygies of the monomial ideal lift to syzygies of every ideal in the stratum. In the case of marked families, lifting syzygies is more difficult because we do not have a term ordering inducing a noetherian polynomial reduction procedure and, therefore, in general there is not an analogue of Buchberger’s criterion. I will focus on the case when the monomial ideal I is strongly stable and I will show how combinatorial properties of strongly stable ideals allow to define a “special” reduction procedure that is again noetherian and that allows to extend the criterion to lift syzygies.

In the second part of the talk, I will present some results obtained using marked families and some ongoing projects related to computation of global equations of the Hilbert scheme, smoothability of zero-dimensional Gorenstein schemes and construction of family of curves with general moduli.

Modifications, faithful tropicalization and moduli spaces of plane tropical curves

by Hannah Markwig (Tübingen)
Friday, 9 December 2016 at 09:30 in Room 230 at the Fields Institute

Tropical geometry can be viewed as an efficient degeneration technique in algebraic geometry, with important applications for instance in enumerative geometry. To make good use of it, it can be necessary to focus on faithful tropicalizations. Modifications provide a concrete tool to construct faithful tropicalizations. We discuss the use and effect of modifications in the case of curves of genus one, two and three. This talk is based on joint work with Maria Angelica Cueto and Ralph Morisson.

Combinatorics and moduli of tropical linear series on curves

by Margarida Melo (Roma Tre)
Monday, 5 December 2016 at 11:00 in Room 230 at the Fields Institute

The theory of linear series on tropical curves, since its introduction by Baker and Norine about 10 years ago, has seen spectacular developments in recent years. In fact, the combinatorial systematic treatment of degenerations of classical linear series that the theory allows has led to the proof of many important results on algebraic curves. On the other hand, the introduction and study of a number of tropical moduli spaces of curves along with its realization as skeletons of their classical (compactified) counterparts allows for a deeper understanding of combinatorial aspects of moduli spaces and their compactifications. In this talk, I will explore this principle for certain moduli spaces of bundles on curves, as the moduli space of spin structures and their compactified/tropical versions.

Classical invariant theory and birational geometry of moduli spaces

by Han-Bom Moon (Fordham)
Wednesday, 7 December 2016 at 09:30 in Room 230 at the Fields Institute

Invariant theory is a study of the invariant subring of a given ring equipped with a linear group action. Describing the invariant subring was one of the central mathematical problems in the 19th century and many great algebraists such as Cayley, Clebsch, Hilbert, and Weyl had contributed to it. There are many interesting connections between invariant theory and modern birational geometry of moduli spaces. In this talk I will explain some concrete examples including the moduli space of parabolic vector bundles on the projective line and the moduli space of stable rational pointed curves. This talk is based on joint work with Swinarski and Yoo.

Moduli of rational and elliptic curves in toric varieties

by Dhruv Ranganathan (MIT)
Thursday, 8 December 2016 at 09:30 in Room 230 at the Fields Institute

The moduli spaces of stable maps to toric varieties occur naturally in the context of enumerative geometry. While they have several excellent properties, they are nonetheless quite singular, reducible, and even non-equidimensional. In genus 0, the situation becomes markedly improved by adding logarithmic structure to the moduli problem. This yields irreducible toroidal compactifications of the space of maps. In turn, tropical geometry gives strong control over the global geometry of this space. For elliptic maps, logarithmic structures alone do not suffice to desingularize these moduli space. However, in conjunction with insights of Abramovich and Wise, the polyhedral geometry of elliptic tropical curves can be used to construct irreducible toroidal compactifications of the moduli space of elliptic curves in toric varieties, generalizing work of Vakil and Zinger. Time permitting, enumerative applications will be discussed.

On the structure of stable cohomology for toroidal compactifications of Ag

by Orsola Tommasi (Chalmers)
Friday, 9 December 2016 at 11:00 in Room 230 at the Fields Institute

Principally polarized abelian varieties of dimension g are basic objects in algebraic geometry, but the cohomology of their moduli space Ag is largely unknown. However, by a classical result of Borel, the cohomology of Ag in degree k < g is is freely generated by the odd Chern classes of the Hodge bundle. Work of Charney and Lee provides an analogous result for the stable cohomology of the minimal compactification of Ag, the Satake compactification. For most geometric applications, it is more natural to consider toroidal compactifications of Ag instead. In this case, we have some stability results for the perfect cone compactification and the matroidal partial compactification. In this talk, we will consider the combinatorial aspects of this stable cohomology and its relationship with the structure of the toroidal fans.

The cohomology of the Hilbert scheme and of the compactified Jacobians of a singular curve

by Filippo Viviani (Roma Tre)
Thursday, 8 December 2016 at 11:00 in Room 230 at the Fields Institute

We generalize the classical MacDonald formula for smooth curves to reduced curves with planar singularities. More precisely, we show that the cohomologies of the Hilbert schemes of points on a such a curve are encoded in the cohomologies of the fine compactified Jacobians of its connected subcurves, via the perverse Leray filtration. A crucial step in the proof is the case of nodal curves, where everything reduces to some combinatorial identities on the underlying dual graph. This is a joint work with Luca Migliorini and Vivek Schende.