Higher syzygies of hyperelliptic curves

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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

ASJC Scopus subject areas

  • Algebra and Number Theory

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