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