Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

# Simple Search

Query:
Enter a query and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server