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Multiplicity bound for theta divisors on indecomposable ppavs

Prove that for every indecomposable principally polarized abelian variety (A, Θ) of dimension g, the multiplicity of the theta divisor at any point is at most (g+1)/2.

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Background

For Jacobians, the multiplicity bound mult_D Θ ≤ (g+1)/2 is classical and stems from the geometry of theta divisors on curves. Extending this bound to all indecomposable principally polarized abelian varieties would provide a powerful global constraint on their theta divisors.

The authors note that this conjecture is proved for g ≤ 5 via the Prym construction but remains open in general, underscoring gaps in our understanding of theta divisor singularities beyond Jacobians and Pryms.

References

As we will see, theta divisors of Jacobians are in general very special and singular, and one can conjecture that this bound for the multiplicity of theta divisors holds for all indecomposable \footnote{A principally polarized abelian variety is called indecomposable if it is not a product of lower-dimensional principally polarized abelian varieties. For such products (called decomposable) the multiplicity of the theta function is the sum of the multiplicities on the factors, and decomposable abelian varieties need to be excluded from many geometric discussions.} $A_\tau\inA_g$. This conjecture is proven for $g\le 5$ in , where the Prym construction provides a geometric description of all abelian varieties, but remains completely open in general, showing how little we understand the geometry of theta divisors of arbitrary abelian varieties.

— Integrable systems approach to the Schottky problem and related questions (2504.20243 - Grushevsky et al., 28 Apr 2025) in Section “Theta functions on Jacobians” (Remark following Exercise 3)