NOETHER INEQUALITY FOR A NEF AND BIG DIVISOR ON A SURFACE

Title & Authors
NOETHER INEQUALITY FOR A NEF AND BIG DIVISOR ON A SURFACE
Shin, Dong-Kwan;

Abstract
For a nef and big divisor D on a smooth projective surface S, the inequality $\small{h^{0}}$(S;$\small{O_{s}(D)}$) $\small{{\leq}\;D^2\;+\;2}$ is well known. For a nef and big canonical divisor KS, there is a better inequality $\small{h^{0}}$(S;$\small{O_{s}(K_s)}$) $\small{{\leq}\;\frac{1}{2}{K_{s}}^{2}\;+\;2}$ which is called the Noether inequality. We investigate an inequality $\small{h^{0}}$(S;$\small{O_{s}(D)}$) $\small{{\leq}\;\frac{1}{2}D^{2}\;+\;2}$ 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;
Language
English
Cited by
References
1.
W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin-Heidelberg-New-York, 1984

2.
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New-York, 1978

3.
M. Kobayashi, On Noether's inequality for threefolds, J. Math. Soc. Japan 44 (1992), no. 1, 145-156

4.
B. Saint-Donat, Projective models of K3 surfaces, Amer. J. of Math. 96 (1974), no. 4, 602-639