REMARKS ON NONSPECIAL LINE BUNDLES ON GENERAL k-GONAL CURVES
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In this work we obtain conditions for nonspecial line bundles on general k-gonal curves failing to be normally generated. Let L be a nonspecial very ample line bundle on a general k-gonal curve X with k ≥ 4 and degL ≥ 3 2 g + g−2 k + 1. If L fails to be normally generated, then L is isomorphic to KX −(ng1k +B)+R for some n ≥ 1, B and R satisfying (1) h0(R) = h0(B) = 1, (2) n+3 ≤ degR ≤ 2n+2, (3) deg(R∩F) ≤ 1 for any F ∈ g1k . Its converse also holds under some additional restrictions. As a corollary, a very ample line bundle L ≃ KX−g0 d+ξ0/e is normally generated if g0/d ∈ X(d) and ξ0/e ∈ X(e) satisfy d ≤ g/2 − g−2/k − 3, supp(g0/d ∩ ξ0/e ) = ∅ and deg(g0/d ∩ F) ≤ k − 2 for any F ∈ g1/k .
Abstract
In this work we obtain conditions for nonspecial line bundles on general k-gonal curves failing to be normally generated. Let L be a nonspecial very ample line bundle on a general k-gonal curve X with k ≥ 4 and degL ≥ 3 2 g + g−2 k + 1. If L fails to be normally generated, then L is isomorphic to KX −(ng1k +B)+R for some n ≥ 1, B and R satisfying (1) h0(R) = h0(B) = 1, (2) n+3 ≤ degR ≤ 2n+2, (3) deg(R∩F) ≤ 1 for any F ∈ g1k . Its converse also holds under some additional restrictions. As a corollary, a very ample line bundle L ≃ KX−g0 d+ξ0/e is normally generated if g0/d ∈ X(d) and ξ0/e ∈ X(e) satisfy d ≤ g/2 − g−2/k − 3, supp(g0/d ∩ ξ0/e ) = ∅ and deg(g0/d ∩ F) ≤ k − 2 for any F ∈ g1/k .
참고문헌 (8)
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