Twisted cubics on cubic fourfolds

TL;DR

This paper constructs a new family of 8-dimensional symplectic manifolds as moduli spaces of twisted cubics on certain cubic fourfolds, revealing new geometric structures and contractions.

Contribution

It introduces a twenty-dimensional family of irreducible holomorphic symplectic manifolds via moduli spaces of twisted cubics on cubic fourfolds, with a novel contraction to a symplectic manifold.

Findings

The moduli space M_3(Y) is smooth and 20-dimensional.

There exists a contraction from M_3(Y) to a projective 8-dimensional symplectic manifold Z(Y).

The construction relies on linear determinantal representations of cubic surfaces.

Abstract

We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M_3(Y) -> Z(Y) to a projective eight-dimensional symplectic manifold Z(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces.