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.
Edoardo Ballico, Alessandra Bernardi (2012). A partial stratification of secant varieties of Veronese varieties via curvilinear subschemes. SARAJEVO JOURNAL OF MATHEMATICS, 8(1), 33-52.
A partial stratification of secant varieties of Veronese varieties via curvilinear subschemes
BERNARDI, ALESSANDRA
2012
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.