# Biregular and birational geometry of quartic double solids with 15 nodes

## TL;DR

This paper classifies certain degree 2 del Pezzo threefolds with 15 nodes, showing they relate to special quartic hypersurfaces and analyzing their automorphisms and birational rigidity.

## Contribution

It proves that del Pezzo varieties with 15 nodes correspond to hyperplane sections of the Igusa quartic and classifies their G-birationally rigid forms.

## Key findings

- Del Pezzo varieties with 15 nodes are linked to hyperplane sections of the Igusa quartic.
- Automorphism groups are derived from the Coble fourfold.
- Classification of G-birationally rigid varieties of this type.

## Abstract

Three-dimensional del Pezzo varieties of degree 2 are double covers of projective space $\mathbb{P}^{3}$ branced in a quadric. In this paper we prove that if a del Pezzo variety of degree 2 has exactly 15 nodes then the corresponding quadric is a hyperplane section of the Igusa quartic or, equivalently, all such del Pezzo varieties are members of one particular linear system on the Coble fourfold. Their automorphism groups are induced from the automorphism group of Coble fourfold. Also we classify all $G$-birationally rigid varieties of such type.