Hyperflex loci of hypersurfaces

TL;DR

This paper investigates the geometric properties of the $k$-flex locus in general hypersurfaces, computing its dimension and degree, and establishing uniqueness and contact order properties of flex lines, using advanced vector bundle techniques.

Contribution

It provides explicit formulas for the dimension and degree of the $k$-flex locus for any $k$, generalizes previous work, and analyzes the structure of flex lines in hypersurfaces.

Findings

Computed the dimension and degree of the $k$-flex locus for general hypersurfaces.

Proved that through a generic $k$-flex point passes a unique $k$-flex line.

Established that the contact order of the flex line is exactly $k$ when $k \

Abstract

The $k$-flex locus of a projective hypersurface $V\subset \mathbb P^n$ is the locus of points $p\in V$ such that there is a line with order of contact at least $k$ with $V$ at $p$. Unexpected contact orders occur when $k\ge n+1$. The case $k=n+1$ is known as the classical flex locus, which has been studied in details in the literature. This paper is dedicated to compute the dimension and the degree of the $k$-flex locus of a general degree $d$ hypersurface for any value of $k$. As a corollary, we compute the dimension and the degree of the biggest ruled subvariety of a general hypersurface. We show moreover that through a generic $k$-flex point passes a unique $k$-flex line and that this line has contact order exactly $k$ if $k\le d$. The proof is based on the computation of the top Chern class of a certain vector bundle of relative principal parts, inspired by and generalizing a work of Eisenbud and Harris.