Moduli spaces of 6 x 6 skew matrices of linear forms on P^4 with a view towards intermediate Jacobians of cubic threefolds

TL;DR

This paper studies the moduli space of 6x6 skew-symmetric matrices of linear forms on P^4, focusing on their relation to cubic threefolds and intermediate Jacobians, with an emphasis on compactification and family constructions.

Contribution

It provides a concrete description of an irreducible component of the incidence correspondence of Pfaffian representations of cubic threefolds.

Findings

Characterization of the irreducible component dominating the space of skew matrices.

Insights into the compactification of Pfaffian representations.

Framework for studying families of cubic threefolds and their Jacobians.

Abstract

It is well known that every smooth cubic threefold is the zero locus of the Pfaffian of a 6 x 6 skew-symmetric matrix of linear forms in P^4. To compactify the space of such Pfaffian representations of a given cubic and to study the construction in families as well as for singular or reducible cubics, it is thus natural to consider the incidence correspondence of Pfaffian representations inside the product of the space of semistable skew-symmetric 6 x 6 matrices of linear forms in P^4 and the space of cubics. Here we describe concretely the irreducible component of this incidence correspondence dominating the space of skew matrices.