## On the geometry of higher secants to rational normal curves

Nivaldo Medeiros
Abstract:
Given a rational normal curve
$C$
in a projective space
${\mathbb P}^{2r}$
, let
$Sec_r(C)$
be the secant of
$r-1$
-planes to
$C$
. Despite being a classical object, there are still many unanswered questions about the geometry and topology of these hypersurfaces.
Our approach consists to study the gradient map
$\mathbb P^{2r} \dashrightarrow \mathbb P^{2r}$
, given by the partial derivatives of the equation defining
$Sec_r(C)$
. We prove that the degree of this map coincides with the degree of the Grassmannian of lines in
$\mathbb P^{r+1}$
. Consequently this map is not birational whenever
$r\geq 2$
, answering in the affirmative a conjecture raised by Maral Mostafazadehfard and Aron Simis.
Now we have a conjecture of our own, namely a generating function for the multidegree of these maps for all
$r$
. This would yield important invariants associated to these secants, such as the Chern-Schwartz-MacPherson class and the Segre class of the singular locus.
Work in progress, joint with Jefferson Nogueira and Giovanni Staglianò.