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General components of the Noether-Lefschetz locus and their density in the space of all surfaces. (English) Zbl 0671.14017
Let $$\Sigma$$ (d) denote the projective space corresponding to surfaces in $${\mathbb{P}}_ 3$$ of degree $$d$$ (d$$\geq 4)$$ and S(d) the Zariski open subset of $$\Sigma$$ (d) corresponding to non-singular surfaces. Let NL(d)$$\subset S(d)$$ consist of those surfaces S with Pic(S) not generated by $${\mathcal O}_ s(1)$$. The Noether-Lasker theorem states that NL(d) is a countable union of proper closed irreducible subvarieties of S(d). Let V be one of these subvarieties. Then, V will be called general if $$p_ G(d)=c(V)$$ where c(V) is the codimension of V in S(d). The author’s main result shows that there are infinitely many such general V and that the union of such general V is dense in S(d) (over any algebraically closed field). Over the complex numbers a proof of density due to Green in respect of the natural topologies is outlined.
Reviewer: P.Cherenack

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14C05 Parametrization (Chow and Hilbert schemes) 14N05 Projective techniques in algebraic geometry
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##### References:
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