The hyperosculating spaces of hypersurfaces
type: Item
instance of: scholarly article
title: The hyperosculating spaces of hypersurfaces
zbMATH Open document ID: 0789.14039
DOI: 10.1007/BF02571447
author: Mingwei Xu
published in: Mathematische Zeitschrift
publication date: 1993-03-07T00:00:00Z
full work available at URL: https://eudml.org/doc/174460
review text: This paper can be looked at as a contribution to the generalization of Weierstrass points and gap sequences to higher dimensions. For a plane curve the problem can be dealt with by studying its inflections. For a smooth hypersurface \(X\) in \(\mathbb{P}^ n\) we are thus led to considering the contact order \(b\) of almost all the tangent hyperplanes and the hyperosculating space, i.e. the scheme of the contact points of the tangent hyperplanes whose contact order is \(>b\). This notion had been studied by the author in two previous papers for \(n=3\). Here that discussion is generalized to any \(n\) when the characteristic of the ground field is \(p \neq 2\). If \(b \neq 2\), then \(p \neq 0\), and \(b\) is shown to be a power of \(p\). This case is characterized by supplying a suitable expression for the polynomial defining \(X\). When the hyperosculating space is finite, it is also studied as a 0-cycle in any characteristic.
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 126414
zbMATH Keywords: inflection
zbMATH Keywords: contact order
zbMATH Keywords: Weierstrass points
zbMATH Keywords: gap sequences
zbMATH Keywords: hypersurface
zbMATH Keywords: hyperosculating space
reviewed by: Marino Palleschi
MaRDI profile type: Publication
cites work: Q3328651
cites work: Algebraic geometry and local differential geometry
cites work: Q4143433
cites work: Q4154669
cites work: Symmetric forms and Weierstrass cycles
cites work: Q4165460
cites work: Q3921342
cites work: Q4775956
cites work: Q3950666
cites work: Wronskians and Plücker formulas for linear systems on curves
cites work: Weierstrass points as singularities of maps in arbitrary characteristic
cites work: On the Points of Weierstrass in Dimensions Greater Than One
cites work: Polar classes of singular varieties
cites work: Q4771357
cites work: Tangency and Duality Over Arbitrary Fields
cites work: Coordinate gap and bidualily
cites work: Q3478544
cites work: Q3475401
OpenAlex ID: W2050402624
description: scientific article; zbMATH DE number 126414