A remark on some geometric invariants over perfect fields
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.
scientific article; zbMATH DE number 3912507