A bound on the Castelnuovo-Mumford regularity for curves

Abstract.

Recall that a projective curve in $\Bbb P^r$ with ideal sheaf $\Cal I$ is said to be n-regular if $H^i(\Bbb P^r, \Cal I\otimes\Cal O_{\Bbb P^r}(n-i))=0$ for every integer $i>0$ and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus $p_a$ satisfies a smaller regularity bound than the optimal one $d-r+2$. For example, if $p_a\geq r-2$ then a curve is $(d-2r+4)$-regular unless it is embedded by a complete linear system of degree $2p_a+2$.