Hilbert Scheme of a Pair of Skew Lines on Cubic Threefolds

TL;DR

This paper investigates the Hilbert scheme component associated with pairs of skew lines on smooth cubic threefolds, establishing its smoothness, structure, and connections to Fano varieties, hyperplane sections, and Bridgeland moduli spaces.

Contribution

It proves the smoothness and describes the structure of the Hilbert scheme component for skew lines on cubic threefolds, linking it to Fano varieties and moduli spaces.

Findings

The component is smooth and isomorphic to a blow-up of a symmetric product.

It reveals geometric relations between lines, singularities, and moduli spaces.

Provides new insights into the structure of line configurations on cubic threefolds.

Abstract

A pair of disjoint lines on a smooth cubic threefold determines an irreducible component of the Hilbert scheme. We prove that this component is smooth and isomorphic to the blow-up of the symmetric product of Fano varieties of lines on the diagonal. We also study its relation to the geometry of lines and singularities on the hyperplane sections and its relation to Bridgeland moduli spaces.