Let ℒ be a line bundle on a smooth curve C of genus g ≥ 2 and let W ⊂ H0(ℒ) be a subspace of dimension r + 1, in this paper we study the natural map μW: W ⊗ H0(ωC) → H0(ℒ⊗ωC). Let D ⊂ G(r + 1, H0(ℒ)) be the locus where μWfails to be surjective: we prove that, if C is not hyperelliptic of genus g ≥ 3, D is an irreducible and reduced divisor on G(r + 1, H0(ℒ)) for any r ≥ 3, and for r = 2 if the curve C is not trigonal.
Brivio, S. (2002). On the degeneracy locus of a map of vector bundles on Grassamannian varieties. MATHEMATISCHE NACHRICHTEN, 244(1), 26-37 [10.1002/1522-2616(200210)244:1<26::AID-MANA26>3.0.CO;2-L].
On the degeneracy locus of a map of vector bundles on Grassamannian varieties
Brivio, S
2002
Abstract
Let ℒ be a line bundle on a smooth curve C of genus g ≥ 2 and let W ⊂ H0(ℒ) be a subspace of dimension r + 1, in this paper we study the natural map μW: W ⊗ H0(ωC) → H0(ℒ⊗ωC). Let D ⊂ G(r + 1, H0(ℒ)) be the locus where μWfails to be surjective: we prove that, if C is not hyperelliptic of genus g ≥ 3, D is an irreducible and reduced divisor on G(r + 1, H0(ℒ)) for any r ≥ 3, and for r = 2 if the curve C is not trigonal.
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