Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes

Details Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes

2010-10-18

arxiv.org/abs/1010.3546v2

Mathematics

Algebraic Geometry

16 pages

Scientific paper

We give a partial "quasi-stratification" of the secant varieties of the order $d$ Veronese variety $X_{m,d}$ of $\mathbb {P}^m$. It covers the set $\sigma_t(X_{m,d})^{\dagger}$ of all points lying on the linear span of curvilinear subschemes of $X_{m,d}$, but two "quasi-strata" may overlap. For low border rank two different "quasi-strata" are disjoint and we compute the symmetric rank of their elements. Our tool is the Hilbert schemes of curvilinear subschemes of Veronese varieties. To get a stratification we attach to each $P\in \sigma_t(X_{m,d})^{\dagger}$ the minimal label of a quasi-stratum containing it.

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