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arXiv · 1702.06204

A generic global Torelli theorem for certain Horikawa surfaces

Gregory Pearlstein·Zheng Zhang
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Abstract

Algebraic surfaces of general type with $q=0$, $p_g=2$ and $K^2=1$ were described by Enriques and then studied in more detail by Horikawa. In this paper we consider a $16$-dimensional family of special Horikawa surfaces which are certain bidouble covers of $\mathbb{P}^2$. The construction is motivated by that of special Kunev surfaces which are counterexamples for infinitesimal Torelli and generic global Torelli problem. The main result of the paper is a generic global Torelli theorem for special Horikawa surfaces. To prove the theorem, we relate the periods of special Horikawa surfaces to the periods of certain lattice polarized $K3$ surfaces using eigenperiod maps and then apply a Torelli type result proved by Laza.

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Gregory Pearlstein, Zheng Zhang. 2017-02-20. A generic global Torelli theorem for certain Horikawa surfaces. https://arxiv.org/abs/1702.06204

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