Double Veronese cones with singularities

TL;DR

This paper investigates the properties of double Veronese cones, a class of three-dimensional algebraic varieties, focusing on singularities, automorphisms, and rationality criteria.

Contribution

It provides sharp bounds on the number of nodes, characterizes automorphism groups, and establishes rationality conditions for these cones.

Findings

Bound of 21 nodes for a specific example

Structure of automorphism groups determined

Criteria for rationality and unirationality established

Abstract

We study double Veronese cones -- three-dimensional del Pezzo varieties of degree one -- with terminal Gorenstein singularities. We prove sharp bounds for the number of nodes, determine the structure of the automorphism group, and establish criteria for rationality and unirationality. In particular, we exhibit a $\mathbb{Q}$-factorial nodal double Veronese cone with $21$ nodes.