# Self-dual matroids from canonical curves

**Abstract**: Self-dual configurations of \(2n\) points in a projective space of dimension \(n-1\) were studied by Coble, Dolgachev–Ortland, and Eisenbud–Popescu. We examine the self-dual matroids and self-dual valuated matroids defined by such configurations, with a focus on those arising from hyperplane sections of canonical curves. These objects are parametrized by the self-dual Grassmannian and its tropicalization. We tabulate all self-dual matroids up to rank five and investigate their realization spaces. Following Bath, Mukai, and Petrakiev, we explore algorithms for recovering a curve from the configuration. A detailed analysis is given for self-dual matroids arising from graph curves.

The code for our project can be found on the git repository <https://github.com/sachihashimoto/self-dual> so that users can easily access the data and track changes to the repository.

This page and the presentation and accessibility of the data are a work in progress. If you have any questions, please email us.

## Small self-dual matroids

In our paper, we study self-dual matroids, i.e. matroids that are equal to their own dual. E.g. in rank 3, the two self-dual matroids on the set [6] are the uniform matroid, with no nonbases, and the matroid with nonbases {123, 456}.

The self-dual matroids up to rank 5 can be found in the collection \(\texttt{Matroids: Self-Dual Matroids}\) in the \(\texttt{polyDB}\) database https://polydb.org.

Additional algebraic data, like realization spaces etc, can be found as described below.

**Data for rank 4**

We see that specializing the template gives the the self-dual configurations of the matroid (avoiding certain bad values like x[14] = 1, which belong to the closure of the self-dual realization space but not he self-dual realization space) subject to the extra constraint that x[7]*x[8]*x[11]-x[7]*x[8]*x[12]-x[7]*x[11]*x[12]+x[7]*x[12]+x[8]*x[11]*x[12]-x[8]*x[11].

**Data for rank 5**

Additionally, for almost all matroids with non-empty self-dual realization space, we obtained example self-dual realizations of the matroid using the code here and variations of it. The sage code here verifies the output. | The point configurations that were defined over the rationals are given in this file and the ones over number fields are in this file.

## Lifting to Canonical Curves

Using the \(\verb|Macaulay2|\) code found here we pass genus 6 curves through the rational point configurations obtained by the self-dual realizations of rank 5 matroids. These genus 6 curves are recorded here.

## The Cayley Octad map

The files here and here give the Maple code for our proof of Theorem 2.6 about the shape of the Cayley Octad map.

## Graph Curves

We also provide data on graph curves and a census of matroids arising from graph curves in small genus. This file contains \(\texttt{Macaulay2}\) code which: inputs a trivalent 3–connected simple graph with genus g and \(2g-2\) vertices; constructs the corresponding graph curve with genus \(g\) and degree \(2g-2\); intersects the graph curve with a random hyperplane; and outputs the self-dual matroid of rank \(g-1\) on \(2g-2\) elements from the resulting configuration of \(2g-2\) points in the hyperplane section. For each considered graph, we ran the algorithm multiple times to make sure the hyperplane was most generic. We considered all such graphs of genus 4 through 7 and we collected the data we obtained here; this data is split by genus (see the following files for genus 4, 5, 6, and 7).

## Self-duality in tropical geometry

We used computations of the tropical Grassmannians \(\text{trop}(\text{Gr}(3,6))\), \(\text{trop}(\text{Gr}(2,6))\) and \(\text{trop}(\text{SGr}(3,6))\) to prove Theorem 6.5. These computations were done using \(\verb'polymake 4.6'\) and \(\verb'Singular 4.3.1'\). All the data and explanations are stored here .

The Data of the tropical Grassmannian \(\text{trop}(\text{Gr}(3,6))\) was originally computed by Speyer and Sturmfels for “The tropical Grassmannian” and analysed by Herrmann, Jensen, Joswig and Sturmfels for the paper “How to draw tropical planes”. It is also available at the internet page https://www.uni-math.gwdg.de/jensen/Research/G3_7/grassmann3_6.html The encoding of the data that we used, as well as the scripts, were provided by Benjamin Schröter in the style used for the Data of the paper “Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians”.

The documents cone0 to cone6, available here, contain the intersections of the 6 representative cones of \(\text{trop}(\text{Gr}(3,6))\) with the space of self-dual points, which coincides with \(\text{trop}(\text{SGr}(3,6))\). The cones are \(\texttt{polymake}\) objects, i.e. the files can be loaded into polymake and investigated there. For this open polymake and type \(\verb|$C0 = load("your/path/to/cone0");|\)

In order to reproduce these data, run this code within \(\texttt{polymake}\). This file calls the data on cones and rays of the tropical Grassmannian \(\text{trop}(\text{Gr}(3,6))\) modulo the group action and scripts to recover \(\text{trop}(\text{SGr}(3,6))\).

The Singular code including output to compute the tropical Variety of \(\text{trop}(\text{Gr}(2,6))\) can be found here.

The tropical Variety of \(\text{trop}(\text{SGr}(3,6))\) was computed using \(\texttt{Singular}\), the input and output of the code is available here.

Project page created: 30/11/2022.

Project contributors: Alheydis Geiger, Sachi Hashimoto, Bernd Sturmfels, Raluca Vlad.

Corresponding author of this page: Sachi Hashimoto, sachi.hashimoto@mis.mpg.de.

Software used: Magma (V2.26-11), Julia (Version 1.8.3), OSCAR (version 0.11.0), SageMath (version 9.7), Python (3.10.6), Polymake (4.6), Singular (4.3.1), Maple, GNU Parallel, Macaulay2 (version 1.19.1)

Last updated 18/09/23.