General components of the Noether-Lefschetz locus and their density in the space of all surfaces
type: Item
instance of: scholarly article
title: General components of the Noether-Lefschetz locus and their density in the space of all surfaces
zbMATH Open document ID: 0671.14017
DOI: 10.1007/BF01462891
author: Ciro Ciliberto
author: Joe Harris
author: Rick Miranda
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164489
review text: 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.
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 4097545
zbMATH Keywords: Noether-Lefschetz locus
zbMATH Keywords: geometric genus
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: surfaces
zbMATH Keywords: of degree d
zbMATH Keywords: Noether-Lasker theorem
MaRDI profile type: Publication
cites work: Q3312351
cites work: A new proof of the explicit Noether-Lefschetz theorem
cites work: On the Noether-Lefschetz theorem and some remarks on codimension-two cycles
OpenAlex ID: W1999028675
description: scientific article; zbMATH DE number 4097545