×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Carlson, J., Green, M., Griffiths, P., Harris, J.: Infinititesimal variations of Hodge structures. Compos. Math.50, 109-205 (1983) · Zbl 0531.14006
[2] Green, M.: A new proof of the explicit Noether-Lefschetz theorem. Preprint · Zbl 0674.14005
[3] Griffiths, P., Harris, J.: On the Noether-Lefschetz theorem and some remarks on codimension two cycles. Math. Ann.271, 31-51 (1985) · Zbl 0552.14011 · doi:10.1007/BF01455794
[4] Lopez, A.: Brown University Ph. D. thesis, 1988
[5] Sernesi, E.: Topics on families of projective schemes. Queen’s Pap. Pure Appl. Math.73 (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.