Mathematics – Algebraic Geometry
Scientific paper
2003-11-27
Trans. Amer. Math. Soc. 358 (2006), no. 2, 759--798
Mathematics
Algebraic Geometry
36 pages. To appear in Transactions of the American Mathematical Society
Scientific paper
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 $\phi$ 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, an alternative description as a double cover of the plane is also given, and the index of the paracanonical system is computed.
No associations
LandOfFree
Surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2 does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2 will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-519257