Involutions on surfaces with $p_g=q=0$ and $K^2=3$

Details Involutions on surfaces with $p_g=q=0$ and $K^2=3$ Involutions on surfaces with $p_g=q=0$ and $K^2=3$

2010-07-28

arxiv.org/abs/1007.5036v1

Mathematics

Algebraic Geometry

13 pages

Scientific paper

We study surfaces of general type $S$ with $p_g=0$ and $K^2=3$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$. It is shown that, if $S/i$ is not rational, then $S/i$ is birational to an Enriques surface or it has Kodaira dimension $1$ and the possibilities for the ramification divisor of the covering map $S\rightarrow S/i$ are described. We also show that these two cases do occur, providing an example. In this example $S$ has a hyperelliptic fibration of genus $3$ and the bicanonical map of $S$ is of degree $2$ onto a rational surface.

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