I read somewhere that the space of rank-one tensors, known as the Segre variety, defined by $$ Seg: \mathbb{P}V_1 \times \cdots \times \mathbb{P}V_n \rightarrow \mathbb{P}(V_1 \otimes \cdots \otimes V_n)$$ $$([v_1],\ldots,[v_n]) \mapsto [v_1 \otimes \cdots \otimes v_n] $$ is a variety. This means that it should be the zero locus of some polynomials. But how does one plug tensors into polynomials?

  • $\begingroup$ Whenever you have a vector space $V$ the homogeneous coordinate ring of $\mathbf P V$ is $\operatorname{Sym} V^*$. Of course if $V \simeq k^n$ then $\operatorname{Sym} V^* \simeq k[x_1,\dots,x_n]$ but some of the elegance is lost. $\endgroup$ – Hoot Sep 24 '15 at 19:54

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