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EnglishWe 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 $\phi$ 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 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, we also give an alternative description as a double cover of the plane.
Surfaces of general type with p_g=q=1, K^2=8 and bicanonical map of degree 2 / Polizzi, Francesco. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - 358(2006), pp. 759-798.
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| Titolo: | Surfaces of general type with p_g=q=1, K^2=8 and bicanonical map of degree 2 |
| Autori: | |
| Data di pubblicazione: | 2006 |
| Rivista: | |
| Citazione: | Surfaces of general type with p_g=q=1, K^2=8 and bicanonical map of degree 2 / Polizzi, Francesco. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - 358(2006), pp. 759-798. |
| Handle: | http://hdl.handle.net/20.500.11770/124959 |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
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