Generalizations of Igusa surfaces

Abstract

We give a criterion for the contracted product $E{\times }^{G}X$ to have a singular Picard scheme, where $E$ is an elliptic curve, $G\subset E$ is a finite subgroup scheme, and $X$ is a projective $G$-normal curve, in the sense of Brion. We show that every smooth projective surface with a singular Picard scheme and a positive-dimensional automorphism group scheme arises in this way. We also illustrate the criterion by constructing two new families of surfaces with singular Picard schemes, arising from $G$-normal curves $X$ for $G=\mathbb{Z}/p\mathbb{Z}$ and $G={\mathbf{\alpha }}_p$, respectively. As a special case of $G=\mathbb{Z}/2\mathbb{Z}$, we recover Igusa's classical example. Finally, we compute all infinitesimal group schemes which appear as ${\mathrm{Pic}}_S^0/({\mathrm{Pic}}_S^0{)}_{\mathrm{red}}$ for all (quasi)-bielliptic surfaces $S$. This completes the description of the singular Picard schemes of smooth projective surfaces of Kodaira dimension zero.

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