Double Veronese cones with 28 nodes

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.

Appendix is made more precise, including the codes. We also added a comment after Theorem 1.4