2023/09/17 by Jefferson Nogueira, Nogueira, Jefferson
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2309.09254
openalex publication_date 2023/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study characteristic classes of hypersurfaces in the complex projective space, with emphasis on secants to rational normal curves. For Seck C⊂ ℙn, the secant of k points to a rational normal curve C⊂ ℙn, we compute the Hilbert series and the topological Euler characteristic. For n=2r and k=r, the case when Secr C⊂ ℙ2r is a hypersurface, we show that the dual (Secr C)^* is isomorphic to the Veronese variety ν2(ℙr), from which we obtain, for Secr C, formulas for the Mather class, the generic Euclidean distance degree and its polar degrees. Furthermore, we present an explicit formula for the topological degree of the gradient map ϕr \colon ℙ2r \dashrightarrow ℙ2r associated with Secr C, and as a consequence we obtain an affirmative answer for a conjecture by M. Mostafazadehfard and A. Simis: for r ≥ 2, the hypersurface Secr C⊂ ℙ2r is not homaloidal. From computations in particular cases we are led to a conjecture, namely, explicit formulas for the projective degrees of the gradient map ϕr and the Schwartz-MacPherson class cSM(Secr C)∈ A_* ℙ2r, for all r. We conclude by presenting evidence that indicates the validity of our conjecture. Keywords: Characteristic classes. Gradient maps. Secants to rational normal curves.