On higher syzygies of ruled surfaces II

In this article we continue the study of property Np of irrational ruled surfaces begun in [E. Park, On higher syzygies of ruled surfaces, math.AG/0401100, Trans. Amer. Math. Soc., in press]. Let X be a ruled surface over a curve of genus g 1 with a minimal section C0 and the numerical invariant e. When X is an elliptic ruled surface with e=−1, it is shown in [F.J. Gallego, B.P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996) 626–659] that there is a smooth elliptic curve E X such that E≡2C0−f. And we prove that if L PicX is in the numerical class of aC0+bf and satisfies property Np, then (C,LC0) and (E,LE) satisfy property Np and hence a+b 3+p and a+2b 3+p. This gives a proof of the relevant part of Gallego–Purnaprajnaʹ conjecture in [F.J. Gallego, B.P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996) 626–659]. When g 2 and e 0 we prove some effective results about property Np. Let L PicX be a line bundle in the numerical class of aC0+bf. Our main result is about the relation between higher syzygies of (X,L) and those of (C,LC) where LC is the restriction of L to C0. In particular, we show the followings: (1) If e g−2 and b−ae 3g−2, then L satisfies property Np if and only if b−ae 2g+1+p. (2) When C is a hyperelliptic curve of genus g 2, L is normally generated if and only if b−ae 2g+1 and normally presented if and only if b−ae 2g+2. Also if e g−2, then L satisfies property Np if and only if a 1 and b−ae 2g+1+p.