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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 ofcurvilinear subschemes of $X_{m,d}$, but two "quasi-strata" may overlap. Forlow border rank two different "quasi-strata" are disjoint and we compute thesymmetric rank of their elements. Our tool is the Hilbert schemes ofcurvilinear subschemes of Veronese varieties. To get a stratification we attachto each $P\in \sigma_t(X_{m,d})^{\dagger}$ the minimal label of a quasi-stratumcontaining it.

Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes