# On higher syzygies of ruled surfaces II

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### Abstract

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 ∈ Pic X 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 ∈ 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, 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.

Original language English 590-608 19 Journal of Algebra 294 2 https://doi.org/10.1016/j.jalgebra.2005.05.022 Published - 2005 Dec 15 Yes

### Fingerprint

Syzygies
Ruled Surface
Elliptic Surfaces
If and only if
Genus
Algebra
Hyperelliptic Curves
Line Bundle
Elliptic Curves
Continue
Restriction
Curve
Invariant

### ASJC Scopus subject areas

• Algebra and Number Theory

### Cite this

In: Journal of Algebra, Vol. 294, No. 2, 15.12.2005, p. 590-608.

Research output: Contribution to journalArticle

Park, Euisung. / On higher syzygies of ruled surfaces II. In: Journal of Algebra. 2005 ; Vol. 294, No. 2. pp. 590-608.
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N2 - 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 ∈ Pic X 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 ∈ 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, 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.

AB - 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 ∈ Pic X 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 ∈ 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, 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.

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