General components of the Noether-Lefschetz locus and their density in the space of all surfaces
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.
scientific article; zbMATH DE number 4097545