Arrondo, Enrique ; Bernardi, Alessandra
(2009)
On the variety parametrizing completely decomposable polynomials.
[Preprint]
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Abstract
The purpose of this paper is to relate
the variety parameterizing completely decomposable
homogeneous polynomials of degree $d$ in $n+1$ variables on
an algebraically closed field, called
$\Split_{d}(\PP n)$, with the Grassmannian of $n-1$
dimensional projective subspaces of $\PP {n+d-1}$. We compute
the dimension of some secant varieties to
$\Split_{d}(\PP n)$ and find a counterexample to a conjecture
that wanted its dimension related to the one of the secant
variety to $\GG (n-1, n+d-1)$. Moreover by using an invariant
embedding of the Veronse variety into the Pl\"ucker space, then
we are able to compute the intersection of $\GG (n-1,
n+d-1)$ with
$\Split_{d}(\PP n)$, some of its secant variety, the tangential
variety and the second osculating space to the Veronese
variety.
Abstract
The purpose of this paper is to relate
the variety parameterizing completely decomposable
homogeneous polynomials of degree $d$ in $n+1$ variables on
an algebraically closed field, called
$\Split_{d}(\PP n)$, with the Grassmannian of $n-1$
dimensional projective subspaces of $\PP {n+d-1}$. We compute
the dimension of some secant varieties to
$\Split_{d}(\PP n)$ and find a counterexample to a conjecture
that wanted its dimension related to the one of the secant
variety to $\GG (n-1, n+d-1)$. Moreover by using an invariant
embedding of the Veronse variety into the Pl\"ucker space, then
we are able to compute the intersection of $\GG (n-1,
n+d-1)$ with
$\Split_{d}(\PP n)$, some of its secant variety, the tangential
variety and the second osculating space to the Veronese
variety.
Document type
Preprint
Creators
Keywords
Decomposable polynomials,
Secant varieties,
Grassmannians,
Veronese varieties
Subjects
DOI
Deposit date
16 Mar 2009
Last modified
16 May 2011 12:10
URI
Other metadata
Document type
Preprint
Creators
Keywords
Decomposable polynomials,
Secant varieties,
Grassmannians,
Veronese varieties
Subjects
DOI
Deposit date
16 Mar 2009
Last modified
16 May 2011 12:10
URI
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