Lowest Degree Spanned Line Bundles on Integral Projective Curves

Abstract

IntroductionLet Y be an integral projective curve defined over an algebraically closed field K and TT : X -• Y its normalization. Set g := p a {Y) and q := p a (X). We always assume g > 4. The dualizing sheaf uiy is locally free if and only if Y is Gorenstein. We have hP{Y,uy) = g. Since g > 3, uxy is spanned by its global sections ([10]). Set L := TT*(coy)/Tors(n*(wy)).Since X is smooth, L is a line bundle. We always assume that Y is not hyperelliptic. Hence u is a birational map ([10], Th. 17). Hence C is an integral non-degenerate curve in P 9_1 and deg(C) = d. Thus C is a partial normalization of Y and q < 7 < g.