Remarks on \(r\)-planes in complete intersections
type: Item
instance of: scholarly article
title: Remarks on \(r\)-planes in complete intersections
zbMATH Open document ID: 1372.14003
author: Chikashi Miyazaki
published in: Tokyo Journal of Mathematics
publication date: 2017-04-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1459367269
review text: Consider smooth complete intersections of dimension \(n\) and multidegree \(d = (d_1 , \dots , d_{N-n} )\) in \(\mathbb P^{N}\). The author is interested in the scheme \(H_r\) of such complete intersections which contain a linear space of dimension \(r\). He notes that if \( n \leq 2r -1 \) then Lefschetz theorem implies that \(H_r = \emptyset \). He then proves for \( n \geq 2r \) that \(H_r \neq \emptyset \) and he computes its dimension. Under the same assumption, he further shows that if a second inequality holds, which is a function of the multidegree \(d\) and of \(r\), then every complete intersection of type \(d\) is covered by \(r\)-linear spaces. The method of proof is based on the study of the tangent spaces to the incidence correspondences which naturally appear in the description of the geometry of the problem.
reviewed by: Alberto Collino
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14M10
zbMATH DE Number: 6701604
zbMATH Keywords: complete intersections
zbMATH Keywords: Hilbert schemes
MaRDI profile type: Publication
OpenAlex ID: W2333144678
DOI: 10.3836/TJM/1459367269
description: scientific article; zbMATH DE number 6701604