On higher syzygies of ruled surfaces II

[摘要] 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, rnath.AG/0401 100, Trans. Amer. Math. Soc., in press]. Let X be a ruled surface over a curve of genus g >= 1 with a minimal section C-0 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 subset of X such that E 2C(0) -f. And we prove that if L epsilon Pic X is in the numerical class of aC(0) + 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 Np. Let L epsilon Pic X be a line bundle in the numerical class of aC(0) + bf. Our main result is about the relation between higher syzygies of M L) and those of (C, LC) where LC is the restriction of L to C-0. 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. (c) 2005 Elsevier Inc. All rights reserved.

[发布日期] 2005-12-15