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 $\mathrm{Split}_{d}(\mathbb{P}^{n})$, with the Grassmannian of $n-1$ dimensional projective subspaces of $\mathbb{P}^{n+d-1}$. We compute the dimension of some secant varieties to $\mathrm{Split}_{d}(\mathbb{P}^{n})$ and find a counterexample to a conjecture that wanted its dimension related to the one of the secant variety to $\mathbb{G} (n-1, n+d-1)$. Moreover by using an invariant embedding of the Veronse variety into the Pl\"ucker space, then we are able to compute the intersection of $\mathbb{G} (n-1, n+d-1)$ with $\mathrm{Split}_{d}(\mathbb{P}^{n})$, some of its secant variety, the tangential variety and the second osculating space to the Veronese variety.

On the variety parameterizing completely decomposable polynomials

BERNARDI, Alessandra
2011-01-01

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 $\mathrm{Split}_{d}(\mathbb{P}^{n})$, with the Grassmannian of $n-1$ dimensional projective subspaces of $\mathbb{P}^{n+d-1}$. We compute the dimension of some secant varieties to $\mathrm{Split}_{d}(\mathbb{P}^{n})$ and find a counterexample to a conjecture that wanted its dimension related to the one of the secant variety to $\mathbb{G} (n-1, n+d-1)$. Moreover by using an invariant embedding of the Veronse variety into the Pl\"ucker space, then we are able to compute the intersection of $\mathbb{G} (n-1, n+d-1)$ with $\mathrm{Split}_{d}(\mathbb{P}^{n})$, some of its secant variety, the tangential variety and the second osculating space to the Veronese variety.
2011
215
3
201
220
http://www.sciencedirect.com/science/article/pii/S0022404910000824
Arrondo E; Bernardi A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/129557
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