On the geometry of higher secants to rational normal curves

Nivaldo Medeiros
Universidade Federal Fluminense
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.