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arXiv · 2606.10884

Coble type hypersurfaces and hyperk\"ahler fourfolds

Benedetta Piroddi·Ángel David Ríos Ortiz·Andrés Rojas·Jieao Song
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Abstract

A classical result, already observed by Coble, asserts that a genus 2 Jacobian can be embedded in $\mathbf P^8$ as the singular locus of a unique cubic hypersurface; similarly, the Kummer of a genus 3 Jacobian is embedded in $\mathbf P^7$ as the singular locus of a unique quartic hypersurface. We present a precise analogue of these results in the context of hyperk\"ahler fourfolds: for the general member in the 20-dimensional locally complete families of polarized hyperk\"ahler fourfolds of $\mathrm{K3}^{[2]}$-type with squares 4 or 6 and divisibility 1, we establish the existence of a unique Coble type hypersurface. As a consequence, we describe several geometric aspects of the corresponding moduli spaces.

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Benedetta Piroddi, Ángel David Ríos Ortiz, Andrés Rojas, Jieao Song. 2026-06-09. Coble type hypersurfaces and hyperk\"ahler fourfolds. https://arxiv.org/abs/2606.10884

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