# The Hirota Variety of a Rational Nodal Curve¶

Claudia Fevola and Yelena Mandelshtam: The Hirota Variety of a Rational Nodal Curve

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

\begin{split}\begin{align*} \phi \;:\; \mathbb{C}^{3g} \quad & \dashrightarrow (\mathbb{C}^*)^{2^g}\times \mathbb{W}\mathbb{P}^{3g-1}\\ (\lambda_1,\dots,\lambda_g,\kappa_1,\kappa_2,\dots,\kappa_{2g}) & \longrightarrow (a_{{\bf c}_1},a_{{\bf c}_2},\dots,a_{{\bf c}_{2^g}},{\bf u}, {\bf v}, {\bf w}) \end{align*}\end{split}

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

$\begin{split}\begin{matrix} u_i = \kappa_{2i-1}-\kappa_{2i},\quad v_i = \kappa_{2i-1}^2- \kappa_{2i}^2, \quad w_i = \kappa_{2i-1}^3-\kappa_{2i}^3, \qquad \text{for all } i=1,\dots, g,\\ a_{\mathbf{c}} = \displaystyle\prod_{\substack{i,j\in I\\i<j}} (\kappa_i - \kappa_j) \displaystyle\prod_{i: c_i = 1} \lambda_i \, \quad \text{ where } I = \{2i: c_i = 0\} \cup \{2i-1: c_i = 1\}, \quad \text{for all } \mathbf{c}\in \mathcal{C}. \end{matrix}\end{split}$

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.