On higher syzygies of ruled surfaces II

In this article we continue the study of property N_p 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 ∈ Pic X is in the numerical class of aC₀ + bf and satisfies property N_p, then (C, L _C0) and (E, L _E) satisfy property N_p 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 N_p. Let L ∈ Pic X 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, L_C) where L_C 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 N_p 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 N_p if and only if a ≥ 1 and b - ae ≥ 2g + 1 + p.