Coble duality for Jacobian Kummer fourfolds

TL;DR

This paper explores dual projective models of Jacobian Kummer fourfolds using theta groups and Coble cubic, introduces a new involution, and studies singularities of secants in algebraic geometry.

Contribution

It establishes a duality between two models of Jacobian Kummer fourfolds and constructs a novel involution on the Hilbert square of a Jacobian surface.

Findings

Established duality between two singular models of Kummer fourfolds.

Constructed a non-natural involution on the Hilbert square of a Jacobian surface.

Analyzed singularities of secants of varieties at identifiable points.

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.