Duality for relative Prymians associated to K3 double covers of del Pezzo surface of degree 2

TL;DR

This paper explores the duality of Lagrangian fibrations related to relative Prym varieties associated with K3 double covers of degree 2 del Pezzo surfaces, revealing a rational involution on their moduli space.

Contribution

It characterizes the dual Lagrangian fibration and identifies the moduli space of such fibrations, establishing a rational involution induced by duality.

Findings

Dual Lagrangian fibrations are characterized explicitly.

A moduli space of these fibrations is identified.

Duality induces a rational involution on the moduli space.

Abstract

Markushevich and Tikhomirov provided a construction of an irreducible symplectic V-manifold of dimension 4, the relative compactified Prym variety of a family of curves with involution, which is a Lagrangian fibration with polarization of type (1,2). We give a characterization of the dual Lagrangian fibration. We also identify the moduli space of Lagrangian fibrations of this type and show that the duality defines a rational involution on it.