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Eliminate extraneous components in theta-constant schemes

Determine whether, for g ≥ 5, augmenting and varying the combinatorially constructed sets of explicit theta-constant equations (as in the Farkas–Grushevsky–Salvati Manni framework) can eliminate all extraneous irreducible components so that their common zero locus equals precisely the Jacobian locus J_g in A_g.

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Background

A recent weak solution provides explicit theta-constant equations whose common zero locus contains the Jacobian locus J_g as an irreducible component for g ≥ 5, but also includes additional components.

The authors remark that expanding these equations through alternative combinatorial choices yields further identities; however, it is unclear if this iterative process can remove all extraneous components, thereby achieving a full Schottky characterization.

References

Moreover, the way $F_i$ are obtained involves combinatorial choices, and by making a different combinatorial choice one can obtain still more explicit equations satisfied by theta constants of Jacobians. It is unclear whether eventually one eliminates all extraneous irreducible components of the common zero set, to obtain a full solution to the Schottky problem in this way.

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