Coble duality for Jacobian Kummer fourfolds
Daniele Agostini, Pietro Beri, Franco Giovenzana, Ángel David Ríos Ortiz
TL;DR
This work investigates extrinsic projective models of generalized Kummer fourfolds arising from Jacobians of genus two curves, by leveraging O'Grady's theta groups and the Coble cubic. It establishes a Coble-type duality between two birational models of Kum$_2(A)$, identifying one model with a contraction to a singular fourfold and the other with a contraction yielding the singular locus of the secant variety Sec$(A)$; these models are related via a polar map to the Coble cubic. A key auxiliary development is a non-natural involution on $A^{[2]}$, together with a global covering of Kum$_2(A)$ by Kum$_1(A)$-like surfaces and a detailed study of Weddle-type phenomena and secant geometry. The results extend the geometry of Kummer fourfolds and connect secant varieties, Weddle surfaces, and moduli-theoretic constructions, suggesting new avenues for higher-dimensional Kummer-type families and theta-function descriptions.
Abstract
We study projective models of generalized Kummer fourfolds via O'Grady's theta groups and the classical Coble cubic. More precisely, we establish a duality between two singular models of the generalized Kummer fourfold of a Jacobian abelian surface. We also give projective models for singular Jacobian Kummer varieties of arbitrary dimension. Along the way, we also construct a first non-natural involution on the Hilbert square of a Jacobian surface. In the appendix, we study singularities of secants of arbitrary varieties at identifiable points, following Choi, Lacini, Park and Sheridan.