Classes características e secantes de curvas racionais normais

We study characteristic classes of hypersurfaces in the complex projective space, with emphasis on secants to rational normal curves. For $Se{c}_{k}C\subset {\mathbb{P}}^n$, 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 $Se{c}_{r}C\subset {\mathbb{P}}^{2r}$ is a hypersurface, we show that the dual $(Se{c}_{r}C{)}^{\ast }$ is isomorphic to the Veronese variety ${\nu }_2({\mathbb{P}}^r)$, from which we obtain, for $Se{c}_{r}C$, 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 ${\phi }_r\text{\hspace{0.17em}⁣}:{\mathbb{P}}^{2r}⇢{\mathbb{P}}^{2r}$ associated with $Se{c}_{r}C$, and as a consequence we obtain an affirmative answer for a conjecture by M. Mostafazadehfard and A. Simis: for $r\ge 2$, the hypersurface $Se{c}_{r}C\subset {\mathbb{P}}^{2r}$ 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 $c_{SM}(Se{c}_{r}C)\in A_{\ast }{\mathbb{P}}^{2r}$, 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.