NOETHER INEQUALITY FOR A NEF AND BIG DIVISOR ON A SURFACE

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For a nef and big divisor D on a smooth projective surface S, the inequality <TEX>$h^{0}$</TEX>(S;<TEX>$O_{s}(D)$</TEX>) <TEX>${\leq}\;D^2\;+\;2$</TEX> is well known. For a nef and big canonical divisor KS, there is a better inequality <TEX>$h^{0}$</TEX>(S;<TEX>$O_{s}(K_s)$</TEX>) <TEX>${\leq}\;\frac{1}{2}{K_{s}}^{2}\;+\;2$</TEX> which is called the Noether inequality. We investigate an inequality <TEX>$h^{0}$</TEX>(S;<TEX>$O_{s}(D)$</TEX>) <TEX>${\leq}\;\frac{1}{2}D^{2}\;+\;2$</TEX> 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.