Volume 83, Nº 3 (2019)

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 varietyof degree $2$ has exactly $15$ nodes, then the corresponding quartic is a hyperplanesection of the Igusa quartic or, equivalently, all such del Pezzovarieties are members of a particular linear system on the Coble fourfold.Their automorphism groups are induced from the automorphism group of theCoble fourfold. We also classify all birationally rigid varieties of thistype.

Let $K$ be an algebraically closed field of characteristic differentfrom $2$, $g$ a positive integer, $f(x)$ a polynomial of degree $2g+1$with coefficients in $K$ and without multiple roots,$\mathcal{C}\colon y^2=f(x)$ the corresponding hyperelliptic curve ofgenus $g$ over $K$, and $J$ its Jacobian. We identify $\mathcal{C}$ withthe image of its canonical embedding in $J$ (the infinite point of$\mathcal{C}$ goes to the identity element of $J$). It is well known thatfor every $\mathfrak{b} \in J(K)$ there are exactly $2^{2g}$ elements$\mathfrak{a}\in J(K)$ such that $2\mathfrak{a}=\mathfrak{b}$. Stollconstructed an algorithm that provides the Mumford representationsof all such $\mathfrak{a}$ in terms of the Mumford representation of$\mathfrak{b}$. The aim of this paper is to give explicit formulaefor the Mumford representations of all such $\mathfrak{a}$ in terms ofthe coordinates $a,b$, where $\mathfrak{b}\in J(K)$ is given by a point$P=(a,b) \in \mathcal{C}(K)\subset J(K)$. We also prove that if $g>1$,then $\mathcal{C}(K)$ does not contain torsion points of ordersbetween $3$ and $2g$.

We show that the bounded derived category of coherent sheaves on a generalEnriques surface can be realized as a semi-orthogonal component in thederived category of a smooth Fano variety with diagonal Hodge diamond.

An extremal curve germ is the analytic germ of a threefold with terminalsingularities along a reduced complete curve admitting a contraction whosefibres have dimension at most one. The aim of the present paper is to reviewthe results concerning contractions whose central fibre is irreducible andcontains only one non-Gorenstein point.