Surfaces of general type with $p_g=q=1,K^2=8$ and bicanonical map of degree 2

Abstract

We classify the minimal algebraic surfaces of general type with $p_g=q=1,K^2=8$ and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, so that if $S$ is such a surface then there exist two smooth curves $C,F$ and a finite group $G$ acting freely on $C\times F$ such that $S=(C\times F)/G$. We describe the $C,F$ and $G$ that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus 3, and the bicanonical map $\varphi$ of $S$ is composed with the involution $\sigma$ induced on $S$ by $\tau \times id:C\times F⟶C\times F$, where $\tau$ is the hyperelliptic involution of $C$. In this way we obtain three families of surfaces with $p_g=q=1,K^2=8$ which yield the first known examples of surfaces with these invariants. We compute their dimension, and we show that they are three smooth and irreducible components of the moduli space $\mathcal{M}$ of surfaces with $p_g=q=1,K^2=8$. For each of these families, an alternative description as a double cover of the plane is also given, and the index of the paracanonical system is computed.

36 pages. To appear in Transactions of the American Mathematical Society

References in corpus (1)

• On the classification of surfaces of general type with non-birational bicanonical map and Du Val Double planes

Cited by in corpus (13)

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• On surfaces of general type with p_g=q=1 isogenous to a product of curves
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• Automorphisms of surfaces of general type with q=1 acting trivially in cohomology
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• Non-simple principally polarised abelian varieties
• Explicit Schoen surfaces
• Involutions on surfaces with $p_g=q=1$
• On surfaces with $p_g=q=1$ and non-ruled bicanonical involution
• Isotrivially fibred surfaces and their numerical invariants
• Surfaces with p_g = 0: Constructions and Moduli spaces, Burniat surfaces and deformations of automorphisms
• On equations of double planes with $p_g=q=1$
• Surfaces with p_g=q=1, K^2=8 and nonbirtional bicanonical map