### Abstract

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 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 ⊂ X such that E ≡ 2C_{0} - f. And we prove that if L ∈ 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 N_{p}. Let L ∈ 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 (X, L) and those of (C, L_{C}) where L_{C} 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.

Original language | English |
---|---|

Pages (from-to) | 590-608 |

Number of pages | 19 |

Journal | Journal of Algebra |

Volume | 294 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Dec 15 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*294*(2), 590-608. https://doi.org/10.1016/j.jalgebra.2005.05.022

**On higher syzygies of ruled surfaces II.** / Park, Euisung.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 294, no. 2, pp. 590-608. https://doi.org/10.1016/j.jalgebra.2005.05.022

}

TY - JOUR

T1 - On higher syzygies of ruled surfaces II

AU - Park, Euisung

PY - 2005/12/15

Y1 - 2005/12/15

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.

UR - http://www.scopus.com/inward/record.url?scp=28244458931&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28244458931&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2005.05.022

DO - 10.1016/j.jalgebra.2005.05.022

M3 - Article

AN - SCOPUS:28244458931

VL - 294

SP - 590

EP - 608

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -