Noether inequality for a nef and big divisor on a surface

Commun. Korean Math. Soc. 2008 Vol. 23, No. 1, 11-18 Printed March 1, 2008

Dong-Kwan Shin Department of Mathematics, Konkuk University, Seoul 143-701, Korea

Abstract : For a nef and big divisor $D$ on a smooth projective surface $S$, the inequality $h^0(S,\mathcal O_S(D))\leq D^2+2$ is well known. For a nef and big canonical divisor $K_S$, there is a better inequality $h^0(S,\mathcal O_S(K_S))\leq \frac{1}{2}{K_S}^2+2$ which is called the Noether inequality. We investigate an inequality $h^0(S,\mathcal O_S(D))\leq \frac{1}{2}D^2+2$ 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