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논문

Normal generation and Clifford index on algebraic curves

https://doi.org/10.1007/s00209-007-0112-9

  • 저자Youngook Choi; Seonja Kim; Young Rock Kim
  • 학술지Mathematische Zeitschrift 257/23
  • 등재유형
  • 게재일자(2007)


For a smooth curve C it is known that a very ample line bundle L">L on C is normally generated if Cliff(L">L ) < Cliff(C) and there exist extremal line bundles L">L (:non-normally generated very ample line bundle with Cliff(L">L ) = Cliff(C)) with h1(L)&#x2264;1">h1(L)1 . However it has been unknown whether there exists an extremal line bundle L">L with h1(L)&#x2265;2">h1(L)2 . In this paper, we prove that for any positive integers (g, c) with g =? 2c +? 5 and c&#x2261;0">c0 (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle L">L with h1(L)=2">h1(L)=2 . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle L">L with h1(L)=2">h1(L)=2 . More generally, if C has a line bundle M">M computing the Clifford index c of C with (3c/2)+3<degM&#x2264;g&#x2212;1">(3c/2)+3<degMg1 , then C has such an extremal line bundle L">L .


For a smooth curve C it is known that a very ample line bundle L">L on C is normally generated if Cliff(L">L ) < Cliff(C) and there exist extremal line bundles L">L (:non-normally generated very ample line bundle with Cliff(L">L ) = Cliff(C)) with h1(L)&#x2264;1">h1(L)1 . However it has been unknown whether there exists an extremal line bundle L">L with h1(L)&#x2265;2">h1(L)2 . In this paper, we prove that for any positive integers (g, c) with g =? 2c +? 5 and c&#x2261;0">c0 (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle L">L with h1(L)=2">h1(L)=2 . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle L">L with h1(L)=2">h1(L)=2 . More generally, if C has a line bundle M">M computing the Clifford index c of C with (3c/2)+3<degM&#x2264;g&#x2212;1">(3c/2)+3<degMg1 , then C has such an extremal line bundle L">L .

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