Classes caracter\'isticas e secantes de curvas racionais normais

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.