Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes

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 ${\sigma }_t(X_{m,d}{)}^{†}$ 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 $P\in {\sigma }_t(X_{m,d}{)}^{†}$ the minimal label of a quasi-stratum containing it.