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Zbl 1036.14019
Flamini, F.; Madonna, C.
Geometric linear normality for nodal curves on some projective surfaces. (English)
[J] Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 4, No. 1, 269-283 (2001). ISSN 0392-4041

A projective variety $X$ contained in $\bbfP^r$ is called linearly normal if the linear series cut out on $X$ by the hyperplanes is complete, i.e. $H^1(X,{\cal I}_X(1))=(0)$. For reduced singular curves $C$ the authors of the paper under review consider also the property of geometric linear normality, introduced by {\it L. Chiantini} and {\it E. Sernesi} [Math. Ann. 307, 41--56 (1997; Zbl 0870.14027)], saying that the normalization map $\widetilde C\to C$ cannot be factorized in a non-degenerate map of $\widetilde C$ to $\bbfP^N$, with $N>r$, followed by a projection.\par The main result is that a nodal curve $C'$ in $\bbfP^r$, lying on a smooth, non-degenerate and linearly normal surface $S$, with $H^1(S,{\cal O}_S(1))=(0)$, is geometrically linearly normal if some numerical conditions are satisfied. These conditions involve the number of nodes of $C'$ and the intersection numbers of $H$, the general hyperplane section of $S$, and $C$, a smooth curve on $S$ such that $C'\in\vert C\vert$. Under the same assumptions, the authors also get that the nodes of $C'$ impose independent conditions on $\vert C-H+K_S\vert$.\par The proof uses the theory of Bogomolov instability for bundles on a surface. Some interesting examples are given to show that the upper bound on the set of nodes of $C'$ is sharp. In these examples the Picard group of $S$ is generated by $H$.
[Emilia Mezzetti (Trieste)]

MSC 2000:: *14J29 Surfaces of general type
14H10 Families, algebraic moduli (curves)

Keywords: linearly normal; nodal curves; Bogomolov unstable

Citations: Zbl 0870.14027

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