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Projective normality of algebraic curves and its application to surfaces. (English) Zbl 1127.14028

Let \(C\) be a smooth curve of genus \(g\) and \(L \in \text{Pic}^d(C)\). \(L\) is said to be normally generated if it is very ample and the natural map \(S^m(H^0(C,L)) \to H^0(C,L^{\otimes m})\) is surjective for all \(m>0\). Here the authors prove that any very ample degree \(d\) \(L\) is normally generated if \(d > \max \{2g+2-4h^1(C,L),2g-(g-1)/6-2h^1(C,L)\}\). Now assume the existence of a degree \(3\) map \(u: C \to C'\) with \(C'\) of genus \(q\). They prove that \(\omega _C(-u^\ast (D))\) is very ample and normally generated if \(D\) is an effective divisor of degree \(z\) on \(C'\) and \(4q < z < (g-1)/6 -2q\}\). They also prove (using hyperplane sections) the projective normality of certain surfaces.

MSC:

14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H50 Plane and space curves
14C20 Divisors, linear systems, invertible sheaves
14J99 Surfaces and higher-dimensional varieties
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References:

[1] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris: Geometry of Algebraic Curves I, Springer, New York, 1985. · Zbl 0559.14017
[2] K. Akahori: Classification of projective surfaces and projective normality , Tsukuba J. Math. 22 (1998), 213–225. · Zbl 0965.14015
[3] A. Beauville: Complex Algebraic Surfaces, Cambridge Univ. Press, Cambridge, 1983. · Zbl 0512.14020
[4] P. Griffiths and J. Harris: Principles of Algebraic Geometry, Wiley-Intersci., New York, 1978. · Zbl 0408.14001
[5] M. Green and R. Lazarsfeld: On the projective normality of complete linear series on an algebraic curve , Invent. Math. 83 (1986), 73–90. · Zbl 0594.14010
[6] R. Hartshorne: Algebraic Geometry, Graduate Text in Math. 52 , Springer, New York, 1977. · Zbl 0367.14001
[7] S. Kim and Y. Kim: Projectively normal embedding of a \(k\)-gonal curve , Comm. Algebra 32 (2004), 187–201. · Zbl 1055.14035
[8] S. Kim and Y. Kim: Normal generation of line bundles on algebraic curves , J. Pure Appl. Algebra 192 (2004), 173–186. · Zbl 1049.14016
[9] H. Lange and G. Martens: Normal generation and presentation of line bundles of low degree on curves , J. Reine Angew. Math. 356 (1985), 1–18. · Zbl 0561.14009
[10] D. Mumford: Varieties defined by quadric equations ; in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Ed. Cremonese, Rome, 1970, 29–100. · Zbl 0198.25801
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