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


Author: Francesco Polizzi
Journal: Trans. Amer. Math. Soc. 358 (2006), 759-798
MSC (2000): Primary 14J29, 14J10, 14H37
DOI: https://doi.org/10.1090/S0002-9947-05-03673-1
Published electronically: March 25, 2005
MathSciNet review: 2177040
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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, i.e. 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 $\phi$ of $S$ is composed with the involution $\sigma$ induced on $S$ by $\tau \times id: C \times F \longrightarrow C \times F$, 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 generically smooth, irreducible components of the moduli space $\mathcal{M}$ of surfaces with $p_g=q=1, \; K^2=8$. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val.


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Additional Information

Francesco Polizzi
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Email: polizzi@mat.uniroma2.it, polizzi@math.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9947-05-03673-1
Keywords: Surfaces of general type, bicanonical map, isotrivial fibrations, Galois coverings
Received by editor(s): November 27, 2003
Received by editor(s) in revised form: March 10, 2004
Published electronically: March 25, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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