2025/02/04 by Cristina Bertone, Bertone, Cristina, Martin Weimann +1 · 1 citation
Mathematics · #14J70 #14N15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2502.02075
openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The k-flex locus of a projective hypersurface V⊂ \mathbb Pn is the locus of points p∈ V such that there is a line with order of contact at least k with V at p. Unexpected contact orders occur when k≥ 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≤ 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.