Lines on projective varieties and applications
type: Item
instance of: scholarly article
title: Lines on projective varieties and applications
zbMATH Open document ID: 1271.14082
author: Francesco Russo
publication date: 2012-12-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.3637
review text: This paper gives a review on the Hilbert scheme \(\mathcal{L}_{x,X}\) of lines contained in a smooth projective variety \(X\subset \mathbb{P}^N\) and passing through a general point \(x\in X\). In particular the author focuses on the relation between the properties of \(\mathcal{L}_{x,X}\) and those of the variety \(X\). In the case of varieties defined by quadratic equations, he proves that \(\mathcal{L}_{x,X}\) coincides with the base locus of the second fundamental form of \(X\) at \(x\). Moreover the behaviour of the Hilbert scheme of lines through a general point is studied with respect to projective extensions. Given a projective variety \(X\subset \mathbb{P}^N\), an extension \(Y\subset\mathbb{P}^{N+1}\) of \(X\) is a variety such that \(X\) can be obtained as hyperplane section of \(Y\). In this case also the Hilbert scheme \(\mathcal{L}_{x,Y}\) is an extension of \(\mathcal{L}_{x,X}\), for \(x\in X\) general. This fact allows the author to describe a sufficient condition for the non-existence of non-trivial extensions via Hilbert schemes and to deduce some applications.
reviewed by: Maria Chiara Brambilla
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14M07
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 6113986
zbMATH Keywords: Hilbert scheme of lines
zbMATH Keywords: quadratic manifolds
zbMATH Keywords: second fundamental form
zbMATH Keywords: projective extensions
MaRDI profile type: Publication
OpenAlex ID: W2087476626
arXiv ID: 1009.3637
cites work: Vanishing Theorems, A Theorem of Severi, and the Equations Defining Projective Varieties
cites work: Higher-dimensional algebraic geometry
cites work: Q3342691
cites work: Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity
cites work: Algebraic geometry and local differential geometry
cites work: Varieties of small codimension in projective space
cites work: Geometry of chains of minimal rational curves
cites work: Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation
cites work: Birationality of the tangent map for minimal rational curves
cites work: Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation
cites work: Characterization of Hermitian symmetric spaces by fundamental forms
cites work: RATIONALITY PROPERTIES OF MANIFOLDS CONTAINING QUASI-LINES
cites work: Varieties with quadratic entry locus, II
cites work: Manifolds covered by lines and the Hartshorne Conjecture for quadratic manifolds
cites work: Q4449111
cites work: Q4859907
cites work: On the infinitesimal rigidity of homogeneous varieties
cites work: Griffiths-Harris rigidity of compact Hermitian symmetric spaces
cites work: Fubini–Griffiths–Harris Rigidity of Homogeneous Varieties
cites work: Projective manifolds with ample tangent bundles
cites work: The red book of varieties and schemes
cites work: Varieties \(n\)-covered by curves of degree \(\delta\)
cites work: Varieties with quadratic entry locus. I
cites work: Lines in \(G/P\)
cites work: Q3140224
cites work: Q4012053
DOI: 10.1007/S12215-011-0072-0
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article