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 h0(S,OS(D)) ≤ D² + 2 is well known. For a nef and big canonical divisor KS, there is a better inequality h0(S,OS(KS)) ≤ ½ KS² + 2 which is called the Noether inequality. We investigate an inequality h0(S,OS(D)) ≤ ½ D² + 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.
Abstract
For a nef and big divisor D on a smooth projective surface S, the inequality h0(S,OS(D)) ≤ D² + 2 is well known. For a nef and big canonical divisor KS, there is a better inequality h0(S,OS(KS)) ≤ ½ KS² + 2 which is called the Noether inequality. We investigate an inequality h0(S,OS(D)) ≤ ½ D² + 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.
참고문헌 (4)
- W. Barth, "Compact Complex Surfaces" (1984), Springer–Verlag
- P. Griffiths, "Principles of Algebraic Geometry" (1978), Wiley
- M. Kobayashi, "On Noether’s inequality for threefolds", J. Math. Soc. Japan, 44(1), pp.145~156 (1992)
- B. Saint-Donat, "Projective models of K3 surfaces", Amer. J. of Math, 96(4), pp.602~639 (1974)