A Partial Stratification of Secant Varieties of Veronese Varieties via Curvilinear Subschemes
DOI:
https://doi.org/10.5644/SJM.08.1.03Keywords:
Symmetric tensor rank, symmetric border rank, secant variety, join, Veronese variety, curvilinear schemes, CANDECOMP/PARAFACAbstract
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.
2000 Mathematics Subject Classification. 14N05, 15A69
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