Double Veronese cones with singularities

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.

26 pages, LaTeX, a few corrections are made