Double Veronese cones with 28 nodes

TL;DR

This paper classifies double Veronese cones with 28 nodes, the maximum for such varieties, linking them to smooth plane quartics and exploring their automorphisms and birational rigidity.

Contribution

It establishes a correspondence between these cones and smooth plane quartics, and analyzes their automorphism groups and birational properties.

Findings

Classified all double Veronese cones with 28 nodes

Established a one-to-one correspondence with smooth plane quartics

Constructed non-conjugate embeddings of S4 into Cremona group

Abstract

We study nodal del Pezzo 3-folds of degree $1$ (also known as double Veronese cones) with $28$ singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane quartics and use this correspondence to study their automorphism groups. As an application, we find all $G$-birationally rigid varieties of this kind, and construct an infinite number of non-conjugate embeddings of the group $\mathfrak{S}_4$ into the space Cremona group.