Oberseminar Kombinatorische Algebraische Geometrie

Study of Real Trigonal Curves and applications

A trigonal curve is a reduced, irreducible curve lying on a uniruled Hirzebruch surface, whose projection induces a three-to-one map to the projective line. In this talk, we will introduce the concept of the dessin d'enfant associated with a real trigonal curve, a type of graph embedding used to study Riemann surfaces and provide combinatorial invariants. We will describe the real properties of the dessins and explain their applications to real algebraic geometry and singularity theory. The first application is the classification of the chambers of the moduli space of real rational plane quintics, an instance of the modern formulation of Hilbert’s 16th problem. The second is the classification of the morsifications of a real semi-quasi-homogeneous singularity of type (3,k).

Clemens Nollau (Tübingen)

Towards Brill-Noether Theory for Spectral Curves

This talk is motivated by the classical study of curves via varieties of linear systems and the recent progress in understanding Brill-Noether Theory for curves with a fixed gonality. We give an approach to these problems for the class of spectral curves. To do this we define spectral curves via characteristic polynomials of Higgs bundles. We present results in two special cases: Canonical double covers and spectral curves over the projective line. In the end we give some outlook on ongoing work with Hitchin's spectral curves.

 Tropical trigonal curves

The moduli of algebraic curves admits a well known stratification by gonality. We would like to study the locus of d-gonal curves, but in the moduli space of tropical curves. We will first review the hyperelliptic case, studied by Melody Chan. In particular, for hyperelliptic curves, the divisorial 2-gonality of the metric graph defining the curve is precisely equivalent to its combinatorial analogue, the 2-gonality of its underlying graph. We will show that, if we allow tropical modifications, then such a relation can be generalized also to the trigonal case. This is a work in progress with Margarida Melo.

Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

In this talk, I would like you to meet Hall-Littlewood-Schubert series, a new class of multivariate generating functions. Their definition features semistandard Young tableaux and polynomials resembling the classical Hall-Littlewood polynomials.

Their intrinsic beauty notwithstanding, Hall-Littlewood-Schubert series have many applications to counting problems in algebra, geometry, and number theory. In my talk the spotlight will be on applications to affine Schubert series. These may be seen as an integral analogue of the Poincare polynomials enumerating the rational points over finite fields of classical Schubert varieties. The latter parametrize subspaces of a given vector space by the intersection dimensions with a fixed flag of reference.

This is joint work with Joshua Maglione. I will explain things from scratch, assuming no familiarity with the advanced technical vocabulary used in this abstract.

Send an email to Daniele Agostini for the Zoom link.

Brill--Noether theory of special curves

Brill--Noether theory studies the maps of curves C to projective spaces. The classical Brill--Noether theorem (established by work of Eisenbud, Fulton, Geiseker, Griffiths, Harris, Lazarsfeld) describes the geometry of this space of maps when C is a general curve. However, the theorem fails for special curves, notably curves that are already equipped with some unexpected map to a projective space. The first case of this is when C is a low-degree cover of the projective line. For general such covers, the Hurwitz--Brill--Noether theorem (joint with E. Larson and I. Vogt) provides a suitable analogue. I'll also present results (joint with S. Vemulapalli) regarding the next natural case: when C is equipped with an embedding in the projective plane.

Qaasim Shafi (Heidelberg)

Tropical refined curve counting and mirror symmetry

An old theorem, due to Mikhalkin, says that the number of rational plane curves of degree d through 3d-1 points is equal to a count of tropical curves (combinatorial objects which are more amenable to computations). There are two natural directions for generalising this result: extending to higher genus curves and allowing for more general conditions than passing through points. I’ll discuss a generalisation which does both, as well as recent work connecting it to mirror symmetry for log Calabi-Yau surfaces. This is joint work with Patrick Kennedy-Hunt and Ajith Urundolil Kumaran. 

Cones of divisors for moduli spaces of polarized hyper-Kahler manifolds

We describe how to compute cones of Noether-Lefschetz divisors on orthogonal modular varieties, with a particular view towards moduli spaces of polarized K3 surfaces and hyperkähler manifolds. We then describe some geometric applications of these cone computations for these moduli spaces. This is a joint work with I. Barros, L. Flapan and B. Williams.

14:00 - 15:00: Frederik Witt (Stuttgart): Introduction to toric geometry

15:00 - 15:30: break

15:30 - 16:30: Jürgen Hausen (Tübingen): Classifying Fano varieties with and without torus action

16:30 - 18:00: Speed talks from PhD students from Stuttgart, Tübingen and Ulm.

Afterwards: Chocolate market in Tübingen

Hilbert schemes of points of singular curves and their combinatorics

The Hilbert scheme of points parametrizes sets of points a the variety counted with multiplicity. In the case of smooth curves their structure is well understood, whereas for singular curves their geometry becomes quite complicated. I will report on a joint work with Ángel David Rios Órtiz about a class of singular curves for which we describe the irreducible components of their Hilbert schemes of points and a particularly nice combinatorial description emerging from them.