Fano schemes of linear spaces on hypersurfaces
type: Item
instance of: scholarly article
title: Fano schemes of linear spaces on hypersurfaces
zbMATH Open document ID: 0904.14024
DOI: 10.1007/BF02677454
author: Adrian Langer
published in: Manuscripta Mathematica
publication date: 1997-09-04T00:00:00Z
full work available at URL: https://eudml.org/doc/156282
review text: Let \(X\) be a generic hypersurface of degree \(d\) in \(\mathbb{P}^n\) and let \(F\) be the Fano scheme of the \(k\)-planes contained in \(X\). The author proves that \(F\) is smooth of dimension \[ \varphi(n,k,d) =(k+1) (n-k)- {d+k \choose k} \] and moreover \(F\) is connected if \(\varphi (n,k,d) >0\). The proof is essentially based on the study of the critical points of the projection \(p:I\to H^0 (\mathbb{P}^n, O(d))\), where \(I=\{(\pi,X) \mid\pi \subset X\}\) is the incidence correspondence and the result follows from an accurate analysis of the dimension of some suitable determinantal varieties.
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 1038284
zbMATH Keywords: generic hypersurface
zbMATH Keywords: Fano scheme
zbMATH Keywords: determinantal varieties
reviewed by: Luciana Picco Botta
MaRDI profile type: Publication
cites work: Fano-varieties of lines on hypersurfaces
cites work: Q4859907
description: scientific article; zbMATH DE number 1038284