Projective manifolds swept out by large dimensional linear spaces
type: Item
instance of: scholarly article
title: Projective manifolds swept out by large dimensional linear spaces
zbMATH Open document ID: 0917.14026
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1999-08-04T00:00:00Z
review text: Main theorem. Let \(X\) be an \(n(\geq 2)\)-dimensional smooth projective variety in a projective space \(\mathbb{P}^N\). Assume that for each point \(x\) in \(X\), there exists an \(m\)-dimensional linear subspace \(P_x\) in \(X\) containing the point \(x\) with \(2m\geq n\). Moreover, suppose that the characteristic of the base field is zero. Then for a general point \(x\) the normal bundle \(N_{P_x/X}\) of \(P_x\) in \(X\) is isomorphic to one of the following: (1) \({\mathcal O}_{\mathbb{P}^m}^{\oplus a}\bigoplus{\mathcal O}_{\mathbb{P}^m} (1)^{\oplus b}\) \((a\) and \(b\) are non-negative integers) (2) \(\Omega_{\mathbb{P}^m} (2)\) \((m\geq 2)\), (3) \(T_{\mathbb{P}^m}(-1)\) \((m\geq 2)\). Moreover, corresponding to the above cases, \(X\) is respectively as follows: (1) a \(\mathbb{P}^{n-a}\)-bundle over an \(a\)-dimensional smooth projective variety \(S\) where a general \(m\)-plane \(P_x\) is in the fiber of the canonical projection \((n\geq a\geq n/2)\); (2) an even-dimensional smooth hyperquadric; (3) the Grassmann variety \(\text{Gr}(m+1,1)\) parametrized by lines in \(\mathbb{P}^{m+1}\) with \(n=2m\), if \(m\) is even.
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 1140471
zbMATH Keywords: large dimensional linear spaces
zbMATH Keywords: flag variety
zbMATH Keywords: normal bundle
author: Ei-ichi Sato
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225105
OpenAlex ID: W2032111431
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DOI: 10.2748/TMJ/1178225105
description: scientific article; zbMATH DE number 1140471