On the uniruled Voisin divisor on the LLSvS variety

TL;DR

This paper investigates the structure of the Voisin divisor on the LLSvS variety associated with a smooth cubic fourfold, revealing its uniruled nature and its relation to the branch divisor of a Voisin map resolution.

Contribution

It demonstrates that the divisor of singular cubic surfaces on the LLSvS variety has two irreducible components, one of which is the uniruled branch divisor of a Voisin map resolution.

Findings

The divisor of singular cubic surfaces on Z has two irreducible components.

One component coincides with the uniruled branch divisor of the Voisin map.

The structure of the Voisin divisor relates to the geometry of singular cubic surfaces.

Abstract

Let $Y$ be a smooth cubic fourfold, $F$ be its Fano variety of lines and $Z$ be its associated LLSvS variety, parametrizing families of twisted cubics and some of their degenerations. In this short note, we show that the divisor of singular cubic surfaces on $Z$ has two irreducible components, one of which coincides with the uniruled branch divisor of a resolution of the Voisin map $F\times F \dashrightarrow Z$.