The hyperosculating spaces of hypersurfaces (Q1204303)

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.