Self-dual matroids from canonical curves

  • Alheydis Geiger (Max Planck Institute for Mathematics in the Sciences)
E1 05 (Leibniz-Saal)


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.

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Daniele Agostini

Max Planck Institute for Mathematics in the Sciences

Christian Lehn

Technische Universität Chemnitz

Rainer Sinn

Universität Leipzig