Higher syzygies of hyperelliptic curves

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)101-111
Number of pages11
JournalJournal of Pure and Applied Algebra
Volume214
Issue number2
DOIs
Publication statusPublished - 2010 Feb 1

Fingerprint

Syzygies
Hyperelliptic Curves
Factorization
Linearly
Graded Betti numbers
Minimal Free Resolution
Line Bundle
Involution
Divides
Genus
Module
Curve

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Higher syzygies of hyperelliptic curves. / Park, Euisung.

In: Journal of Pure and Applied Algebra, Vol. 214, No. 2, 01.02.2010, p. 101-111.

Research output: Contribution to journalArticle

@article{3d77eea8f627400f9c901fee330e579e,
title = "Higher syzygies of hyperelliptic curves",
abstract = "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.",
author = "Euisung Park",
year = "2010",
month = "2",
day = "1",
doi = "10.1016/j.jpaa.2009.04.006",
language = "English",
volume = "214",
pages = "101--111",
journal = "Journal of Pure and Applied Algebra",
issn = "0022-4049",
publisher = "Elsevier",
number = "2",

}

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 -