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Zbl 0611.14031
Xiao, Gang
Finitude de l'application bicanonique des surfaces de type général. (Finiteness of the bicanonical map of surfaces of general type). (French)
[J] Bull. Soc. Math. Fr. 113, 23-51 (1985). ISSN 0037-9484

Let S be a complex surface of general type. The rational map associated to the bicanonical system $\vert 2K\vert$ is called the bicanonical map. It is known that in case $K\sp 2\ge 5$, it is a morphism (Bombieri, Reider) and that in case $K\sp 2\ge 10$, it is birational (Bombieri, Francia). In this paper, it is proved that the bicanonical map is generically finite except for the case: $p\sb g=0$, $K\sp 2=1.$ \par The proof goes as follows: Under the hypothesis that the bicanonical map is not generically finite, the author first shows that $p\sb g=0$ and that the bicanonical map gives a genus $2$ fibration over ${\bbfP}\sp 1$ unless $K\sp 2=1$. Secondly, he shows that if $p\sb g=0$ and if S has a genus $2$ fibration, then $K\sp 2\le 2$. Finally, he proves that if $p\sb g=0$, $K\sp 2=2$, then the bicanonical map does not give a genus $2$ fibration. He also gives an example of a surface of general type with $p\sb g=0$, $K\sp 2=2$, which has a genus $2$ fibration over ${\bbfP}\sp 1$. For further study on genus $2$ fibration on a surface, see the author's book, "Surfaces fibrés en courbes de genre deux", Lect. Notes Math. 1137 (1985; Zbl 0579.14028).
[F.Sakai]

MSC 2000:: *14J10 Families, algebraic moduli, classification (surfaces)
14D99 Families, fibrations

Keywords: bicanonical map; surface of general type

Citations: Zbl 0579.14028

Cited in: Zbl 1093.14056 Zbl 1078.14054 Zbl 1073.14056

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