ABSTRACT

Identifiability holds for the k-secant variety σ_k(X) of an embedded variety $X\subset {\mathbb{P}}^r$ if a general q ∈ σ_k(X) is in the linear span of a unique subset of X with cardinality k. Identifiability is true if the general tangential k-contact locus Γ_k ⊂ X has dimension 0. Under certain assumptions on $\dim {\sigma }_x\left(X\right)$ for some specific x>k, we get $\dim {\mathrm{\Gamma }}_k=0$. Here we give some conditions which exclude the case Γ_x a hypersurface for some x>k and get $\dim {\mathrm{\Gamma }}_k=0$ and hence identifiability for the k-secant variety. As an example we consider the case of Segre-Veronese embeddings of multiprojective spaces, in which the elements of ${\mathbb{P}}^r$ corresponds to partially symmetric tensors.

摘要

可识别性适用于嵌入多样性的k -正割多样性σ _k ( X )$X\subset {\mathbb{P}}^r$如果一般q  ∈  σ _k ( X ) 在X的唯一子集的线性范围内，基数为k。如果一般切向k -接触轨迹 Γ _k  ⊂  X的维数为 0，则可识别性为真。在某些假设下$\mathrm{暗淡}{\sigma }_X\left(X\right)$对于某些特定的x > k，我们得到$\mathrm{暗淡}{\mathrm{\Gamma }}_k=0$. 在这里，我们给出一些条件，这些条件排除了 Γ _{x 的}超曲面对于某些x > k的情况，并得到$\mathrm{暗淡}{\mathrm{\Gamma }}_k=0$因此k -正割品种的可识别性。作为一个例子，我们考虑多投影空间的 Segre-Veronese 嵌入的情况，其中的元素${\mathbb{P}}^r$对应于部分对称张量。