# The Hirota Variety of a Rational Nodal CurveΒΆ

The Hirota variety parameterizes solutions to the KP equation arising from a degenerate Riemann theta function. In this work we study in detail the Hirota variety arising from a rational nodal curve of genus \(g\). Of particular interest is the irreducible subvariety defined as the image of the parameterization map

where the coordinates \({\bf a} = (a_{{\bf c}_1}, a_{{\bf c}_2},\dots,a_{{\bf c}_{2^g}})\) are indexed by the points in \(\mathcal{C}=\{0,1\}^g\). The image of \(\phi\) is defined as follows

We call the closure of the image of \(\phi\) the \(\textbf{main component}\) of the Hirota variety and denote it by \(\mathcal{H}_{\mathcal{C}}^M\). Proving that this is an irreducible component of the Hirota variety corresponds to solving a \(\textbf{weak Schottky problem}\) for rational nodal curves. We solve this problem up to genus 9 using computational tools. The proof of this is mainly computational. Our code, implemented in \(\verb|Macaulay2|\), shows that the Jacobian matrix of the Hirota variety \(\mathcal{H}_{\mathcal{C}}\) evaluated ad a general point in the image of the parametrization has the desired rank. This shows that the parametrization map is dominant into the main component \(\mathcal{H}_{\mathcal{C}}^M\).

We include the code computing the image of the Abel map of an irreducible rational nodal curve. This computation justifies the choice of the parameterization for the parameters \(a_i\) in the map \(\phi\) above. This is also implemented in \(\verb|Macaulay2|\).

Project page created: 28/02/2022

Project contributors: Claudia Fevola and Yelena Mandelshtam

Software used: Macaulay2 (Version 1.13)

System setup used: MacBook Pro with macOS Monterey 12.0.1, Processor 2,7 GHz Intel Core i7, Memory 16 GB 2133 MHz LPDDR3, Graphics Intel Iris Graphics 550 1536 MB.

Corresponding author of this page: Claudia Fevola, claudia.fevola@mis.mpg.de