Projective normality of ruled surfaces

Abstract

In this article we study normal generation of irrational ruled surfaces. When C is a smooth curve of genus g, Green and Lazarsfeld proved that a very ample line bundle L is an element of PicX with deg(L) >= 2g+1-2h(1)(X, L)-Cliff(X) is normally generated where Cliff(C) denotes the Clifford index of the curve C (Green and Lazarsfeld, 1986). We generalize this to line bundles on a ruled surface over C.