Components of maximal dimension in the Noether-Lefschetz locus
type: Item
instance of: scholarly article
title: Components of maximal dimension in the Noether-Lefschetz locus
zbMATH Open document ID: 0674.14003
published in: Journal of Differential Geometry
publication date: 1989-01-01T00:00:00Z
review text: Let Y be the Hilbert scheme of surfaces of degree d in \({\mathbb{P}}^ 3\) and let \(\Sigma_ d\) be the subscheme of Y parametrizing smooth surfaces S such that Pic(S) is not generated by \({\mathcal O}_ S(1)\). For any irreducible component \(\Sigma\) of \(\Sigma_ d\), it is known that \(d-3\leq co\dim_ Y\Sigma \leq h^{2,0}(S)=\left( \begin{matrix} d-1\\ 3\end{matrix} \right)\). The main result of this paper states that, if \(co\dim (\Sigma)=d-3\geq 2\), then \(\Sigma\) is the family of surfaces containing a line. The proof is based on a result of the author [J. Differ. Geom. 27, No.1, 155-159 (1988; see the review after the following one)] and a technique of \textit{G. Gotzmann} [Math. Z. 158, 61-70 (1978; Zbl 0352.13009)].
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 4103221
zbMATH Keywords: Hilbert scheme
Wikidata QID: Q115182813
author: Mark Green
reviewed by: Takao Fujita
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442876
OpenAlex ID: W1592780840
DOI: 10.4310/JDG/1214442876
description: scientific article; zbMATH DE number 4103221