Classes caracter\'isticas e secantes de curvas racionais normais

TL;DR

This paper investigates the characteristic classes and secant varieties of rational normal curves in complex projective spaces, providing explicit formulas and confirming conjectures about their geometric and topological properties.

Contribution

It offers new explicit formulas for characteristic classes, Euler characteristics, and degrees related to secants of rational normal curves, and confirms a conjecture on the non-homaloidality of certain hypersurfaces.

Findings

Computed Hilbert series and Euler characteristic of secant varieties.

Proved the dual of secant hypersurfaces is a Veronese variety.

Confirmed that certain secant hypersurfaces are not homaloidal.

Abstract

We study characteristic classes of hypersurfaces in the complex projective space, with emphasis on secants to rational normal curves. For $Sec_k C\subset \mathbb{P}^{n}$, the secant of $k$ points to a rational normal curve $C\subset \mathbb{P}^n$, we compute the Hilbert series and the topological Euler characteristic. For $n=2r$ and $k=r$, the case when $Sec_r C\subset \mathbb{P}^{2r}$ is a hypersurface, we show that the dual $(Sec_r C)^*$ is isomorphic to the Veronese variety $\nu_2(\mathbb{P}^r)$, from which we obtain, for $Sec_r C$, 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 $\phi_r \colon \mathbb{P}^{2r} \dashrightarrow \mathbb{P}^{2r} $ associated with $Sec_r C$, and as a consequence we obtain an affirmative answer for a conjecture by M. Mostafazadehfard and A. Simis: for $r \geq 2$, the hypersurface $Sec_r C\subset \mathbb{P}^{2r}$ 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 $\phi_r$ and the Schwartz-MacPherson class $c_{SM}(Sec_r C)\in A_* \mathbb{P}^{2r}$, 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.