Quadrics on Gushel-Mukai varieties

TL;DR

This paper investigates the Hilbert schemes of quadrics on Gushel-Mukai varieties by connecting them to relative Hilbert schemes of linear subspaces within a related family of quadrics, revealing geometric structures of these varieties.

Contribution

It introduces a novel relationship between quadrics on Gushel-Mukai varieties and linear subspaces in associated quadric families, expanding understanding of their geometric properties.

Findings

Characterization of Hilbert schemes of quadrics on GM varieties

Relation between quadrics on GM varieties and linear subspaces in quadric families

Structural insights into the geometry of GM varieties

Abstract

We study Hilbert schemes of quadrics of dimension $k \in \{0,1,2,3\}$ on smooth Gushel-Mukai varieties $X$ of dimension $n \in \{2,3,4,5,6\}$ by relating them to the relative Hilbert schemes of linear subspaces of dimension $k + 1$ of a certain family, naturally associated with $X$, of quadrics of dimension $n - 1$ over the blowup of $\mathbf{P}^5$ at a point.