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,L|C0) and (E,L|E) satisfy property Np and hence a+b⩾3+p and a+2b⩾3+p. This gives a proof of the relevant part of Gallego-Purnaprajna' 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.