The indeterminacy locus of the Voisin map

TL;DR

This paper investigates the indeterminacy locus of a rational map between hyperk"ahler varieties associated with cubic fourfolds, revealing it corresponds to intersecting lines on the fourfold.

Contribution

It identifies the indeterminacy locus of Voisin's map as the set of intersecting lines, clarifying the map's geometric behavior.

Findings

Indeterminacy locus is the set of intersecting lines.

The map's indeterminacy is explicitly characterized.

Provides insight into the geometry of hyperk"ahler varieties.

Abstract

Beauville and Donagi proved that the variety of lines $F(Y)$ of a smooth cubic fourfold $Y$ is a hyperk\"ahler variety. Recently, C. Lehn, M.Lehn, Sorger and van Straten proved that one can naturally associate a hyperK\"ahler variety $Z(Y)$ to the variety of twisted cubics on $Y$. Then, Voisin defined a degree 6 rational map $\psi:F(Y)\times F(Y)\dashrightarrow Z(Y)$. We will show that the indeterminacy locus of $\psi$ is the locus of intersecting lines.