In this talk we will discuss irreducible holomorphic symplectic manifolds and their singular analogues, irreducible symplectic varieties (ISVs). Our main goal will be to construct examples of ISVs following the strategy outlined in an article by Markushevich and Tikhomirov. We start by fixing a K3 surface S with an antisymplectic involution i. For a choice of a smooth ample curve C on the quotient S/i, one can construct the corresponding compactified relative Prym variety. By the work of Arbarello, Saccà and Ferretti, we know that under certain assumptions if S/i is an Enriques surface, then P is an ISV. Inspired by their result, we investigate the situation when S/i is a rational surface and find sufficient conditions to ensure that P is an ISV. This is a joint work in progress with E. Brakkee, C. Camere, A. Grossi, L. Pertusi and G. Saccà.
