A remark on some geometric invariants over perfect fields
type: Item
instance of: scholarly article
title: A remark on some geometric invariants over perfect fields
zbMATH Open document ID: 0572.14001
DOI: 10.1016/0021-8693(85)90113-9
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Given \(F_s\), a form of degree \(s\) in \(k[X_0,\ldots, X_N]\), let \(C\) denote the induced homogeneous subscheme, \(C\subset k^{n+1}\). There is a subspace \(S\subset C\) maximal with the condition \(C+S=C\) in the sense of addition in \(k^{n+1}\). If \(k\) is a field of characteristic zero, the ideal defining \(S\) can be obtained as a Jacobian extension of the ideal \(<F_s>\). If \(k\) is a perfect field of a positive characteristic, \(S\) is replaced by a flag of subspaces \(S_0\supset\cdots\supset S_r\) such that \(S_ r\subset C\) and is maximal with the condition \(S_r+C=C\). If \(\bar S_0\supset\cdots\supset \bar S_r\subset \bar C\) denote the induced subschemes in \(P^n_k\), the points \(p\) of \(\bar S_i\) are characterized in terms of the degree of inseparability of the projection \(\pi: \bar C\cup (P^n_k-p)\to H\) to a generic hyperplane \(H\subset P^n_k\). If \(I\) denotes a proper ideal of the local ring \(C<X_1,\ldots,X_n>\), Mather has characterized the Thom-Boardman sequence associated to \(I\) in terms of ranks of ideals obtained from \(I\) by Jacobian extensions. A partial extension of this construction is given for a proper ideal \(I\subset kX_1,\ldots,X_n\) where \(k\) denotes a perfect field of positive characteristic.
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 3912507
zbMATH Keywords: ideal in polynomial ring
zbMATH Keywords: ideal in power series ring
zbMATH Keywords: Jacobian extension of the ideal
zbMATH Keywords: flag of subspaces
zbMATH Keywords: positive characteristic
author: Orlando E. Villamayor
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90113-9
OpenAlex ID: W2136395382
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Vertices and polarizations for homogeneous polynomials
cites work: An extension to fields of positive characteristic of Mather's construction of the Thom-Boardman sequence
cites work: Q3900184
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description: scientific article; zbMATH DE number 3912507