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

We give a partial “quasi-stratification” of the secant varieties of the order d Veronese variety X m,d of P m . It covers the set σ 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 ∈ σ t (X m,d ) † the minimal label of a quasi-stratum containing it.