A simple tensor in $V^{\otimes n}$ is one that can be written as $v_1 \otimes \cdots \otimes v_n$ for some choice of $v_i \in V$, these are also called rank 1 tensors. The space of these simple tensors $V^{\otimes n}_{\text{rank} = 1}$ is not closed under addition, but it is closed under scaling - so it forms a well-defined subset of the projective space $\mathbb{CP}(\text{dim}(V)^n-1)$
- Does this space have a name I can look up?
- Is this a projective variety that has some kind of alternate description? What is it's dimension?
For example, in the case $\text{dim}(V) = n = 2$, with a basis of $V$ given by $e_1, e_2$, then one patch of the projective space of all tensors can be parametrised by $$v_1 \otimes v_1 + a_{12} v_1 \otimes v_2 + a_{21} v_2 \otimes v_1 + a_{22} v_2 \otimes v_2$$ and the simple tensors can be parametrised by $$( v_1 + a_2 v_2) \otimes ( v_1 + b_2 v_2) = v_1 \otimes v_1 + b_2 v_1 \otimes v_2 + a_2 v_2 \otimes v_1 + a_2 b_2 v_2 \otimes v_2$$ Then, thought of as a variety, this is cut out by the equation $a_{12} a_{21} = a_{22}$, so it would seem the simple tensors in this case form a codimension-1 variety in the projective space of all tensors.
