### Abstract

Let X be a hyperelliptic curve of arithmetic genus g and let f : X → P^{1} be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d ≤ 2 g. Note that the minimal free resolution of X of degree ≥ 2 g + 1 is already completely known. Let A = f^{*} O_{P1} (1), and let L be a very ample line bundle on X of degree d ≤ 2 g. For m = max {t ∈ Z {divides} H^{0} (X, L ⊗ A^{- t}) ≠ 0}, we call the pair (m, d - 2 m)the factorization type ofL . Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by | L | are precisely determined by the factorization type of L.

Original language | English |
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Pages (from-to) | 101-111 |

Number of pages | 11 |

Journal | Journal of Pure and Applied Algebra |

Volume | 214 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Feb 1 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Higher syzygies of hyperelliptic curves.** / Park, Euisung.

Research output: Contribution to journal › Article

*Journal of Pure and Applied Algebra*, vol. 214, no. 2, pp. 101-111. https://doi.org/10.1016/j.jpaa.2009.04.006

}

TY - JOUR

T1 - Higher syzygies of hyperelliptic curves

AU - Park, Euisung

PY - 2010/2/1

Y1 - 2010/2/1

N2 - Let X be a hyperelliptic curve of arithmetic genus g and let f : X → P1 be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d ≤ 2 g. Note that the minimal free resolution of X of degree ≥ 2 g + 1 is already completely known. Let A = f* OP1 (1), and let L be a very ample line bundle on X of degree d ≤ 2 g. For m = max {t ∈ Z {divides} H0 (X, L ⊗ A- t) ≠ 0}, we call the pair (m, d - 2 m)the factorization type ofL . Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by | L | are precisely determined by the factorization type of L.

AB - Let X be a hyperelliptic curve of arithmetic genus g and let f : X → P1 be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d ≤ 2 g. Note that the minimal free resolution of X of degree ≥ 2 g + 1 is already completely known. Let A = f* OP1 (1), and let L be a very ample line bundle on X of degree d ≤ 2 g. For m = max {t ∈ Z {divides} H0 (X, L ⊗ A- t) ≠ 0}, we call the pair (m, d - 2 m)the factorization type ofL . Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by | L | are precisely determined by the factorization type of L.

UR - http://www.scopus.com/inward/record.url?scp=70349841331&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349841331&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2009.04.006

DO - 10.1016/j.jpaa.2009.04.006

M3 - Article

AN - SCOPUS:70349841331

VL - 214

SP - 101

EP - 111

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 2

ER -