Hyperflex loci of hypersurfaces
Cristina Bertone, Martin Weimann
TL;DR
This work extends the classical study of flex loci to higher contact orders by introducing and analyzing the $k$-flex locus $V_k$ for general degree $d$ hypersurfaces $V\subset \mathbb{P}^n$. The authors develop an incidence framework via $\Gamma_k$ and its projection to the incidence base $\Phi$, then tie the geometry to a vector bundle of relative principal parts $\mathcal{E}^{k-1}(d)$ on $\Phi$. They prove precise dimensional results for $V_k$ and provide an explicit degree formula $\deg(V_k)=N_k(n,d)$ computed from the top Chern class $c_k(\mathcal{E}^{k-1}(d))$, using Schubert calculus on $\mathbb{G}(1,n)$ and Catalan trapezoids to obtain closed expressions. The results recover the classical flex locus case $k=n+1$ (BDSW) and describe the largest ruled subvariety $V_{\infty}$, including its nonemptiness criteria and explicit line counts in low-degree scenarios. Overall, the paper delivers a complete framework for higher-order contact loci in hypersurfaces, combining incidence geometry, Chern-class computations, and explicit combinatorial tools to yield exact dimensional and degree formulas.
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.