# Hilbert squares of K3 surfaces and Debarre-Voisin varieties

## TL;DR

This paper investigates degenerations of Debarre-Voisin hyperk"ahler fourfolds, showing they can specialize to Hilbert squares of K3 surfaces through certain degenerations of trivectors.

## Contribution

It demonstrates that under specific degenerations, Debarre-Voisin varieties become birationally equivalent to Hilbert squares of K3 surfaces, revealing new links between these hyperk"ahler manifolds.

## Key findings

- Debarre-Voisin varieties degenerate to Hilbert squares of K3 surfaces
- Degenerations involve reducible or excessive dimension varieties
- Specializations occur along general 1-parameter degenerations

## Abstract

The Debarre-Voisin hyperk\"ahler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.