Schedule

The Demazure product on integer permutations is an associative operation that can be understood as a greedy multiplication in Bruhat order. In this talk, I will describe some surprising uses of this combinatorial construction for degeneration arguments in Brill--Noether theory. Roughly speaking, the Demazure product provides a mild generalization of the compatibility conditions for limit linear series. I will describe how this point of view unifies some arguments of classical Brill--Noether theory with the newer Hurwitz--Brill--Noether theory, and discuss some open problems.

The moduli space of non-singular curves of genus g admits a natural stratification by gonality, defined as the smallest d for which there is a degree d morphism to the projective line or equivalently a linear series of degree d and rank one.  We focus on the tropical analogue for trigonal curves (d=3), where, unlike the classical case, the different notions of gonality do not always coincide. We prove that for 3-edge-connected tropical curves, these notions agree up to tropical modifications, allowing the construction of the moduli space of 3-edge-connected tropical trigonal curves as a generalised cone complex, with an explicit description of its maximal cells. In the general case, when 3-edge-connectivity might not hold, the notions diverge, but nonetheless we can characterise exactly when this occurs and prove that the resulting spaces share the same topology. Finally, we relate these constructions to the moduli space of tropical admissible degree 3 covers constructed by Cavalieri, Markwig, and Ranganathan, which allows us to deduce further results on the topology of these spaces. This talk is based on a series of works in collaboration with Margarida Melo.

The classical Brill–Noether theorem states that every nondegenerate degree d map from a general curve C of genus g to projective space is a point of expected dimension in the moduli space of such maps. In this talk, I will present an analogous statement for maps from C to smooth projective toric surfaces. I will then discuss the construction of the space of complete quasimaps to Bl_{P^s}^r, obtained as a suitable blow-up of the quasimap space of Ciocan-Fontanine–Kim–Maulik. This space provides an expected-dimension compactification of the moduli space of maps, in a fixed curve class, from C to X. Conjecturally, the insertion of tautological subschemes corresponding to geometric insertions is transverse, lies in the locus of nondegenerate maps, and preserves the expected dimension. Using the Brill–Noether result for toric surfaces mentioned above, the conjecture is verified in dimension 2. 

Unlike moduli space of curves, moduli spaces of abelian varieties do not have one most obvious compactification. There are many compactifications with different advantages and disadvantages. I will try to explain aspects of the cohomology of toroidal compactifications that are independent of choices. I will also give some applications to the cohomology of the moduli space of abelian varieties itself. This is joint work with Dan Petersen and Olivier Taïbi.

Mikhalkin’s “tropical correspondence formula” expresses counts of genus 0 curves in P^2 through point conditions as weighted sums of tropical curves. Much work has been successful at generalizing to arbitrary genus in dimension 2, or higher dimension with genus 0. We present a full tropical correspondence for both logarithmic Gromov—Witten and enumerative curve counting in the first major unsolved case: genus 1 in smooth toric threefolds. This includes joint work with A. Cela.

Abelian surfaces are complex tori whose enumerative invariants satisfy remarkable regularity properties. The computation of their (reduced) Gromov-Witten invariants for the so called primitive classes is fairly well understood and many complete computations are available. A few years ago, G. Oberdieck conjectured a multiple cover formula expressing in a very simple way the invariants for the non-primitive classes in terms of the primitive one. The proof of the conjecture would solve completely the GW theory of abelian surfaces. In this talk, we'll sketch a proof of multiple cover formula conjecture for many insertions. The argument relies on a reduced degeneration formula and on the computation of correlated Gromov-Witten invariants for trivial bundles on elliptic curves.  This is joint work with T. Blomme.

The moduli space of abelian varieties admits a tautological ring generated by lambda classes. In contrast to the space of stable curves, this tautological ring has a remarkably simple presentation. This allows for the construction of a canonical “tautological projection” which maps the full Chow ring onto the tautological subring, providing a left inverse to the inclusion of the latter in the former. Given a Chow class on the moduli space of abelian varieties, it is natural to attempt to compute its tautological projection. For the Torelli locus (the locus of Jacobians) an algorithm was provided by Faber, the difficult step being integrating monomials in lambda classes on the space of stable curves. We establish a corresponding algorithm for the locus of Prym varieties. The novel geometric content is: a closed formula relating three different types of lambda classes (source, target, and Prym) on the moduli space of admissible covers, proved using Grothendieck-Riemann-Roch; a fundamental invariance property of these lambda classes. We have implemented the algorithm on a computer. It is (currently) effective up to genus 10, giving closed formulae for the tautological projection of the Prym locus. This is joint work in progress with Yoav Len and Sam Molcho.

We develop a novel approach to the Brill–Noether theory of curves endowed with a degree k cover of the projective line, via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill–Noether theory on elliptic K3 surfaces via the notion of Bridgeland stability type for objects in their derived category. As a main application, we show that curves on elliptic K3 surfaces serve as the first known examples of smooth k-gonal curves which are general from the viewpoint of Hurwitz–Brill–Noether theory.  In particular, we provide new proofs of the main non-existence and existence results in Hurwitz–Brill–Noether theory. Finally, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz–Brill–Noether theory. Joint work with Soheyla Feyzbakhsh and Andres Rojas. 

Mgn admits many natural modular finite coverings, obtained as spaces of roots of line bundles inside the universal Picard group. Examples include spaces of spin structures, and the torsion of the universal Jacobian. These coverings have well-behaved logarithmic compactifications, parametrizing degenerations of roots. I will explain how to compute the tropicalizations of these compactifications, and express the tropicalizations as a tropical moduli spaces. This involves finding collections of geometrically meaningful discrete invariants for degenerate roots, and constructing deformations between all roots with the same invariants.

The Torelli morphism assigning to a smooth projective curve its Jacobian is well known to be injective. Generalizing to Prym varieties, the situation becomes much richer. Indeed, the Prym-Torelli morphism assigning to an étale double cover of smooth curves its Prym variety is never injective. Donagi introduced the tetragonal construction and gained beautiful insight into this structure, which is particularly striking in genus 6. 

In this talk, I will present a purely tropical analogue of Donagi's construction and main theorem. This is joint work with Thomas Saillez and draws from previous joint work with Dmitry Zakharov.