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Schottky characterization from theta-constant equations for g ≥ 5

Ascertain whether, for every genus g ≥ 5, the full collection of explicit equations in theta constants obtained via Schottky’s approach (including Schottky–Jung proportionalities and related constructions) suffices to characterize the Jacobian locus J_g inside A_g, i.e., whether the common zero locus of all such equations coincides with J_g.

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Background

Schottky’s classical method yields concrete polynomial relations in theta constants that vanish on Jacobians. In genus 4, these lead to explicit characterizations, but for higher genus the situation is much less clear.

The authors explicitly state that it is unknown whether aggregating all equations obtainable by this approach is enough to cut out the Jacobian locus for g ≥ 5, emphasizing that a definitive resolution remains open despite recent weak solutions identifying J_g as a component of a larger zero locus.

References

From Schottky's original approach one can obtain numerous explicit equations satisfied by theta constants of Jacobians, and one can ask whether the full set of equations thus obtained solves the Schottky problem, i.e.~characterizes $J_g\subseteq A_g$. While this question remains completely open for any $g\ge 5$, a weaker result was recently obtained.

— Integrable systems approach to the Schottky problem and related questions (2504.20243 - Grushevsky et al., 28 Apr 2025) in Section “The Schottky problem”