On blowing-up of polarized surfaces

Koji YOKOYAMA

doi:10.2969/jmsj/05130523

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July, 1999 On blowing-up of polarized surfaces

Koji YOKOYAMA

J. Math. Soc. Japan 51(3): 523-533 (July, 1999). DOI: 10.2969/jmsj/05130523

Abstract

Let $(S,L)$ be a polarized surface and let $\pi$ : $\tilde{S}\rightarrow S$ be the blow-up at $r$ points $p_{1},$ . . . , $p_{r}$ on $S$ . Set $\displaystyle \tilde{L}=\pi^{*}L-\sum a_{i}E_{i}$ , where $a_{i}' \mathrm{s}$ are positive integers and $E_{i}' \mathrm{s}$ are the (-1)-curves over $p_{i}$ . We consider whether $\tilde{L}$ is ample or not if $p_{i}' \mathrm{s}$ are in a general position. The cases of sectional genus two are studied especially precisely.