The Riemann-Schottky problem is the problem of determining which principally polarized abelian varieties (PPAV) arise as Jacobian of curves. Riemann showed that the theta divisor on the Jacobian of a hyperelliptic curve has singularity of codimension three. Beauville conjectured that any irreducible PPAV whose theta divisors have singularity of codimension three actually come from hyperelliptic curves. In this talk, I will discuss a refinement of this problem and a partial solution. To achieve this, we develop a theory of higher multiplier ideals for Q-divisors using the theory of complex Hodge modules of Sabbah-Schnell, building on M.Saito’s theory of rational Hodge modules. This is joint work with Christian Schnell.
Kapranov's compactification of the moduli space of $n$ lines in the projective plane gives an example of a moduli space of stable surfaces: it carries a family of certain reducible degenerations of the plane with $n$ "broken lines". For $n=6$, Luxton proved that this compactification is tropical and that it is associated to the tropical Grassmannian of Speyer and Sturmfels. In this talk we consider a dual perspective and construct a compact moduli space parametrizing $n$-pointed degenerations of the plane arising from Mustafin varieties, which was originally proposed by Gerritzen and Piwek. Moreover, for $n=6$ we show that this compactification is tropical and associated to a specific refinement of the tropical Grassmannian.This is joint work with Jenia Tevelev.
Non-hyperelliptic curves of genus g can be canonically embedded in $g-1$ dimensional projective space. A hyperplane section of such a smooth canonical curve is a canonical divisor, consisting of $2g-2$ points. From work by Dolgachev and Ortland it is known that point configurations obtained in such a way are self-associated. We interpret this notion in terms of matroids: A generic hyperplane section of a canonical curve gives rise to an identically self-dual matroid. This can also be seen as a combinatorial shadow of the Riemann-Roch theorem. This procedure also works for graph curves, which are stable canonical curves consisting of $2g-2$ lines that are determined by their intersection behavior. Self-dual point configurations are parametrized by a subvariety of the Grassmannian $Gr(n,2n)$ and its tropicalization. We provide an extensive analysis of the tropical self-dual Grassmannian $trop(SGr(3,6))$ and pave the way for a thorough investigation of tropical Cayley Octads. Further, we classify identically self-dual matroids of rank up to five and determine the dimension of their (self-dual) realization spaces. Building on work by Petrakiev, we investigate the question, which self-dual point configurations can be obtained by a hyperplane section with a canonical curve.
This project is work in progress with Sachi Hashimoto, Bernd Sturmfels and Raluca Vlad.
The k-th Terracini locus of a variety X measures the degeneracy of the canonical map between the abstract k-th secant variety and the k-th secant variety of X.
After introducing this locus, we focus on the case of Segre varieties and we determine the second and third Terracini locus of any Segre variety. Lastly, we explain how a better knowledge of the Terracini locus is essential when dealing with algorithms computing tensor rank decomposition.
The moduli space of hyperbolic structures on a given closed topological surface of genus g is well known - due to the uniformization theorem it is just the moduli space of Riemann surfaces. It is a 6g-6 dimensional orbifold. It has a smooth manifold cover which is homeomorphic to a vector space. The moduli space of convex real projective structures has many similiar properties. It can also be realized as the quotient of a vector space, now of dimension 16g-16. I will describe the moduli space of convex real projective structures, coordinates on it, as well as explicit ways, how one can move around in this moduli space.
Participants
Daniele Agostini
Max Planck Institute for Mathematics in the Sciences
Barbara Betti
Max Planck Institute for Mathematics in the Sciences
Henry Dakin
TU Chemnitz
Nicholas Early
Max Planck Institute for Mathematics in the Sciences
Alheydis Geiger
Max Planck Institute for Mathematics in the Sciences
Mario Kummer
TU Dresden
Christian Lehn
Technische Universität Chemnitz
Emeryck Marie
Technische Universität Chemnitz
Stefano Mereta
Max Planck Institute for Mathematics in the Sciences
Marta Panizzut
MPI MiS
Angel David Rios Ortiz
MPI
Pierpaola Santarsiero
Max Planck Institute for Mathematics in the Sciences
Luca Schaffler
Università Roma Tre
Javier Sendra Arranz
Max Planck Institute for Mathematics in the Sciences
Christian Sevenheck
TU Chemnitz
Elima Shehu
The Max Planck Institute for Mathematics in the Sciences & Osnabrück University
Rainer Sinn
Universität Leipzig
Bernd Sturmfels
MPI Leipzig
Anna Wienhard
Max Planck Institute for Mathematics in the Sciences
Ruijie Yang
Max Planck Institute for Mathematics, Bonn
Claudia Yun
Max Planck Institute for Mathematics in the Sciences
Scientific Organizers
Daniele Agostini
Max Planck Institute for Mathematics in the Sciences
Christian Lehn
Technische Universität Chemnitz
Rainer Sinn
Universität Leipzig
Administrative Contact
Saskia Gutzschebauch
Max Planck Institute for Mathematics in the Sciences
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