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.


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
Full Text: arXiv Euclid


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