Fachbereich Mathematik

Oberseminar Kombinatorische Algebraische Geometrie

Wintersemester 2023/2024

The seminar usually takes place on Wednesday from 10 to 12 (c.t.) in room C5H41/S08.

25.10 Javier Sendra-Arranz (Tübingen)
The Hessian correspondence of hypersurfaces of degree 3 and 4. Let X be a hypersurface in a n-dimensional projective space. The Hessian map is a rational map from X to the projective space of symmetric matrices that sends a point p to the Hessian matrix of the defining polynomial of X evaluated at p. The Hessian correspondence is the map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure of its image via the Hessian map. We study this map for the cases of hypersurfaces of degree 3 and 4. We prove that, for degree 3 and 4 the Hessian correspondence is birational, apart from degree 3 and d=1, where it is of degree two. Moreover, we provide effective algorithms for recovering a hypersurface from its Hessian variety, for degree 3 and any n, and for degree 4 and n even.
26.10 Angel Rios (Paris Saclay) 15 - 16:30, Hankel-Zimmer.
Hodge theory and Hyperkahler manifolds.K3 surfaces are special types of algebraic surfaces that play a central role in several areas of mathematics, including algebraic geometry, arithmetic geometry, and mathematical physics. The celebrated Torelli theorem of Pyatetskii-Shapiro-Shafarevich-Burns-Rapoport  states that two K3 surfaces can be distinguished by their weight-two Hodge structure together with the intersection product; henceforth a K3 surface can be effectively encoded by discrete data, bringing a particularly rich connection with the theory of lattices. For the higher dimensional analogues of K3 surfaces, that is Hyperkahler manifolds, there is still a Torelli theorem, but with several caveats. In this talk I will speak about how to exploit this theorem in order to decide the "shape" of a given Hyperkahler manifold. This is joint work with Benedetta Pirodi (Milano).

Enrico Savi (Nice) 

The Q-algebraicity problem in real algebraic geometry here the abstract

15.11 Luigi Lombardi (Milano Statale)

On the invariance of Hodge numbers of irregular varieties under derived equivalence. A conjecture of Orlov predicts the invariance of the Hodge numbers of a smooth projective complex variety under derived equivalence. For instance this has been verified to the case of varieties of general type. In this talk, I will examine the case of varieties that are fibered by varieties of general type through the Albanese map. For this class of varieties I will prove the derived invariance of Hodge numbers of type h^{0,p}, together with a few other invariants arising from the Albanese map. This talk is based on a joint work with F. Caucci and G. Pareschi.

29.11 Mima Stanojkovski (Trento)

Ferrers diagram rank-metric codes and a conjecture of Etzion and Silberstein. Ferrers diagram rank-metric codes were introduced by Etzion and Silberstein in 2009 as linear spaces of matrices defined over a finite field, whose nonzero elements are supported on a given Ferrers diagram and have rank lower bounded by a fixed positive integer d. In the same work, Etzion and Silberstein proposed a conjecture on the largest dimension of a Ferrers diagram rank-metric code in terms of the defining parameters. Since stated, the conjecture has been verified in a number of cases, often requiring additional constraints on the field size or on the minimum rank d in dependence of the corresponding Ferrers diagram. As of today, this conjecture still remains widely open. I will report on joint work with Alessandro Neri and on our proof of the Etzion-Silberstein conjecture for the classes of strictly monotone and MDS-constructible Ferrers diagrams, without any additional restrictions.


SWAG (Südwestdeutsche Algebraische Geometrie): joint Oberseminar with Stuttgart and Ulm, N15

14-14:45 Kletus Stern, Stability of hypersurfaces, buildings and Berkovich spaces
14:45-15:15 break
15:15-16 Luca Remke, Immaculate loci and exceptional sequences for toric varieties
16-16:15 break
16:15-17 Victoria Schleis, Tropical quiver Grassmannians
17:30-? chocolate market


Parisa Ebrahimian (Tübingen)

On the generalization of tropical enumerative invariants of F_2

Maxim Lvovich Kontsevich found a recursive formula for counting the number of complex plane rational curves of degree d that pass through general given points. These numbers are GW invariants for the plane. Hannah Markwig and Andreas Gathmann provided a completely combinatorial proof of Kontsevich’s formula by using tropical geometric methods. Tropical geometry is a development in algebraic geometry that aims to simplify algebro-geometric problems into purely combinatorial ones and it is used as a modern tool that can solve enumerative problems effectively. To successfully apply tropical geometry to an enumerative problem, a so-called correspondence theorem is required. In this talk, we are going to see a further generalization of Kontsevich’s formula to the case where the curves are in a Hirzebruch surface and they are required to satisfy not only point and line conditions but also multiple cross-ratio conditions.


Mario Kummer (Dresden), Andres Jaramillo Puentes (Essen)

Naive A^1-knot theory
We present an A^1-count of secant lines to a space curve from a joint work with Daniele Agostini and explain how it can be thought of as an arithmetic knot invariant. Building on this we make some first steps towards a naive A^1-knot theory and conclude with some open questions.

Tropical A^1-Enumerative Geometry

Mikhalkin's correspondence theorem establishes a correspondence between algebraic curves on a toric surface and tropical curves. This translates the question of counting the number of algebraic curves through a given number of points to the question of counting tropical curves, i.e. certain graphs, with a given notion of multiplicity through a given number of points which can be solved combinatorially.

In this talk we will present a version of Mikhalkin's correspondence theorem for an arbitrary base field for k-rational points in a joint work with Sabrina Pauli and we will discuss the combinatorial properties of the tropical counts in a joint work with Hannah Markwig, Sabrina Pauli and Felix Röhrle.


Gurvan Mével (Nantes)

Universal polynomials for coefficients of tropical refined invariant in genus 0

In enumerative geometry, some numbers of curves on surfaces are known to behave polynomially when the cogenus is fixed and the linear system varies, whereas it grows more than exponentially fast when the genus is fixed. In the first case, Göttsche's conjecture expresses the generating series of these numbers in terms of universal polynomials.

Tropical refined invariants are polynomials resulting of a weird way of counting curves, but linked with the previous enumerations. When the genus is fixed, Brugallé and Jaramillo-Puentes proved that some coefficients of these polynomials behave polynomially, bringing back a Göttsche's conjecture in a dual and refined setting. In this talk we will investigate the existence of universal polynomials for these coefficients.


24.1 Jules Chenal (Lille)

On the Maximality of T-Surfaces.

O. Viro's patchworking theorem describes how to glue real algebraic hypersurfaces together in order to obtain new algebraic hypersurfaces. In its most simple version (primitive patchwork) the theorem provides us with an algorithm that construct real hypersurfaces from combinatorial data. More precisely, the combinatorial data (called a sign distribution) will produce both a CW-complex and a real hypersurface in a toric variety that are homeomorphic through an isotopy of the ambient toric variety. The Renaudineau-Shaw spectral sequence is a central tool in the study of their homology. It allowed, for instance, for upper bounds of the individual Betti numbers of such hypersurfaces (such inequalities are a refinement of the Smith-Thom inequality). We will present our generalisation of B. Haas' theorem to the case of T-Surfaces and give a combinatorial condition for the T-Surface to be maximal in the sense of the Smith-Thom inequality.

30.1. Yassine El-Maazouz (Aachen). Start at 14:15 in Room C4H33

Tropical invariants of binary quintics and reduction types of Picard curves.

We give a general framework for tropical invariants associated with group actions on arbitrary varieties. This is then applied to find tropical invariants for binary forms by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification of M_{0,n}. This allows us to express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. This is joint work with Paul Helminck and Enis Kaya.

31.1 Yeongrak Kim (Busan), Manoel Jarra (Groningen). Start at 10:00.

Y. Kim: Recursive Koszul flattening and tensor ranks of determinant/permanent tensors

The rank of a tensor T is the minimum number of decomposable tensors whose sum equals to T which extends the notion of the matrix rank. Understanding the rank of a given tensor has great theoretical and practical applications, however, the rank of a tensor of high order is very hard to determine in most cases. For instance, Strassen's algorithm for matrix multiplication tells us that we only need 7 multiplications (not 8) when we multiply two 2 by 2 matrices, in other words, the 2 by 2 matrix multiplication tensor has rank 7. Usually, the study of rank complexities of a tensor is based on a flattening method that derives a certain matrix from the given tensor. The Koszul flattening method, introduced by Landsberg and Ottaviani, is a simple and powerful method that works for a tensor of order 3 using the exterior product. It has several applications in the study of lower bounds of tensor ranks and Waring ranks for various tensors (of order 3) appearing in algebra and geometry, including the matrix multiplication tensor and the determinant/permanent polynomial for 3 by 3 matrices. 

Motivated by their observations, I will introduce a recursive Koszul flattening method, a successive usage of Koszul flattening for tensors of higher orders. As applications, I will discuss some observations on the lower bounds on tensor ranks of the determinant and permanent as tensors of order n. These results greatly improve lower bounds on the border ranks of those tensors for n at least 4. This is a joint work in progress with Jong In Han and Jeong-Hoon Ju. 

M. Jarra: Category of matroids with coefficients

Matroids are combinatorial abstractions of the concept of
independence in linear algebra. There is a way back: when representing a
matroid over a field we get a linear subspace. Another algebraic object
for which we can represent matroids is the semifield of tropical
numbers, which gives us valuated matroids. In this talk we introduce
Baker-Bowler's theory of matroids with coefficients, which recovers both
classical and valuated matroids, as well linear subspaces, and we show
how to give a categorical treatment to these objects that respects
matroidal constructions, as minors and duality. This is a joint work
with Oliver Lorscheid and Eduardo Vital.


Angelina Zheng (Roma Tre): Stable cohomology of moduli spaces of hyperelliptic curves on Hirzebruch surfaces

In this talk we present the stable cohomology of moduli spaces of genus g hyperelliptic curves, embedded in a Hirzebruch surface. This is done using two different methods. The first one is Gorinov-Vassiliev’s method, which computes the cohomology of complements of discriminants. Similar information on the cohomology of a moduli space can be obtained through point counting over finite fields.
We also describe these moduli spaces of embedded hyperelliptic curves in terms of moduli spaces of pointed non-embedded hyperelliptic curves.
This is a joint work with Jonas Bergström.

Sommersemester 2023

19.4. Giulia Iezzi

26.4. Felix Röhrle

10.5. Levin Stuke

25.5. Patience Ablett 

7.6. Carl Lian

14.6. Jürgen Hausen

27.6. Isabel Vogt

5.7. Türkü Celik

6.7. Benjamin Schroeter