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A new proof of the explicit Noether-Lefschetz theorem. (English) Zbl 0674.14005
Let $$Y:=\{algebraic$$ surfaces of degree d in $${\mathbb{P}}_ 3\}$$ and $$\Sigma_ d:=\{S\in Y| \quad S\quad is\quad$$ smooth and Pic(S) is not generated by the hyperplane bundle$$\}$$. Previously, the author proved the explicit Noether-Lefschetz theorem [J. Differ. Geom. 20, 279-289 (1984; Zbl 0559.14009)]: For $$d\geq 3$$, every component of $$\Sigma_ d$$ has codimension $$\geq d-3$$ in Y. Here the author gives a new and short proof of this result as a consequence of some vanishing theorem for Koszul cohomology on $${\mathbb{P}}_ n$$, the proof of which is given in this paper.
Reviewer: Vo Van Tan

##### MSC:
 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14C22 Picard groups 14C05 Parametrization (Chow and Hilbert schemes) 14F20 Étale and other Grothendieck topologies and (co)homologies
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