On complex surfaces with 5 or 6 semistable singular fibers over P^1

Details On complex surfaces with 5 or 6 semistable singular fibers over P^1 On complex surfaces with 5 or 6 semistable singular fibers over P^1

2004-01-15

arxiv.org/abs/math/0401190v3

Mathematics

Algebraic Geometry

AMS Tex file, 12 pages. This new version is based in previous preprints by S.-L. Tan and Y.Tu and A. G. Zamora

Scientific paper

Let $f:X@>>>\Bbb P^1$ be a fibered surface with fibers of genus g>1. If f is semistable and non isotrivial we prove that X of non negative Kodaira dimension implies that the number s of singular fibers is at least 5. Information about the nature of X if s=6,5 and g<6 is given. If f is any relatively minimal fibration we bound by below K_f^2, the bound depending on g and the Kodaira dimension of X, a classification of rational surfaces with minimal K_f^2 is given. Moreover, we prove several properties of positivity for the linear system K_X+F (F a fibre of f). Examples of a K3 surfaces admitting a semistable fibration with s=6 and g=3 and of a surface of general type admitting a semistable fibration with s=7 and g=4 are provided.

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