NOETHER INEQUALITY FOR A NEF AND BIG DIVISOR ON A SURFACE

Shin, Dong-Kwan

• Published : 2008.01.31
• 49 5

Abstract

For a nef and big divisor D on a smooth projective surface S, the inequality $h^{0}$(S;$O_{s}(D)$) ${\leq}\;D^2\;+\;2$ is well known. For a nef and big canonical divisor KS, there is a better inequality $h^{0}$(S;$O_{s}(K_s)$) ${\leq}\;\frac{1}{2}{K_{s}}^{2}\;+\;2$ which is called the Noether inequality. We investigate an inequality $h^{0}$(S;$O_{s}(D)$) ${\leq}\;\frac{1}{2}D^{2}\;+\;2$ like Clifford theorem in the case of a curve. We show that this inequality holds except some cases. We show the existence of a counter example for this inequality. We prove also the base-locus freeness of the linear system in the exceptional cases.

Keywords

linear system;Noether inequality;nef and big divisor

References

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4. W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin-Heidelberg-New-York, 1984

Cited by

1. Geography of Irregular Gorenstein 3–folds vol.67, pp.03, 2015, https://doi.org/10.4153/CJM-2014-033-0