Self-dual matroids from canonical curves

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.