• Journal of the Korean Mathematical Society
• /
• v.42 no.2
• /
• pp.335-351
• /
• 2005

This paper concerns the relation between the degree and the projective dimension of linear systems on curves. We generalize Clifford's theorem and its improvement by Coppens and G. Martens and classify the special curves for our problem, and estimate their gonality.

• Communications of the Korean Mathematical Society
• /
• v.23 no.1
• /
• pp.11-18
• /
• 2008

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.