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. |