On Higher Syzygies of Ruled Surfaces Ii

In this article we we continue the study of property Np of irrational ruled surfaces begun in [12]. 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, there is an 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 [5]. 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.