A partial stratification of secant varieties of Veronese varieties via curvilinear subschemes

Edoardo Ballico (1) , Alessandra Bernardi (2)

1 UNITN - Università degli Studi di Trento = University of Trento
2 GALAAD - Geometry, algebra, algorithms

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.

Mots clés: symmetric tensor rank; symmetric border rank; secant variety; join; Veronese variety; curvilinear schemes; CANDECOMP/PARAFAC

Domaines: Géométrie algébrique [math.AG]

Edoardo Ballico, Alessandra Bernardi. A partial stratification of secant varieties of Veronese varieties via curvilinear subschemes. 2011. ⟨inria-00610522v1⟩