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The weak Schottky problem for Hirota varieties of rational nodal curves

Let gg be a genus, and let HC\mathcal{H}_\mathcal{C} be the Hirota variety associated with the rational nodal curve data, with main component HCM\mathcal{H}_\mathcal{C}^{M} and the parametric representation referred to as

. **Weak Schottky problem.** For any genus $g$, the main component $\mathcal{H}_\mathcal{C}^{M}$ is a $3g$-dimensional irreducible component of $\mathcal{H}_\mathcal{C}$ with a parametric representation given by

. This conjecture would generalize the corresponding result for rational nodal curves to every genus and provide a solution to the weak Schottky problem; the analogous statement for smooth curves was proven by Dubrovin.

Sources & referencesView supporting material

Primary source

Claudia Fevola and Yelena Mandelshtam, “Hirota Varieties and Rational Nodal Curves”, arXiv:2203.00203 (2023).

algebraic-geometryintegrable-systemsschottky-problem

Progress summary

Refreshed
Partially solved

The conjecture is proved only through genus 9, while the statement for all genera remains open.

The weak Schottky conjecture asserts that the parametrized main component is an irreducible component of the Hirota variety of dimension 3g3g for every genus gg. A 2022 preprint establishes the parametrization and its birationality, but formulates the all-genus assertion as a conjecture.

Known results

  • The parametrization is birational onto an irreducible image of dimension 3g3g.
  • For g≤9g \le 9, the main component is an irreducible component of the Hirota variety, verified by a Macaulay2 Jacobian-rank computation.
  • The stronger equality with the distinguished component remains conjectural.

Current status (as of August 2026): The weak Schottky assertion is established for g≤9g \le 9; no all-genus proof or counterexample has been reported, so the conjecture remains open for g≥10g \ge 10.

Sources

Solutions 0

No solutions have been posted yet.