# Most tensor subspaces of low dimension have rank-1 defining equations

MathOverflow Asked on January 5, 2022

Let $$V_1,ldots , V_k$$ be vector spaces of dimensions $$n_1,ldots , n_k$$ over a field of characteristic zero.

Consider the rational map
$$newcommand{PP}{mathbb{P}} newcommand{bs}{boldsymbol} DeclareMathOperator{Gr}{Gr} DeclareMathOperator{im}{im}$$
$$Phi : left[PP (V_1^*) times cdots times PP (V_k^*)right]^j dashrightarrow Gr (n_1 cdots n_k – j, , V_1otimes cdots otimes V_k )$$
defined by
$$left([alpha_1^i], ldots , [alpha_k^i]right)_{1le ile j} mapsto { T mid (alpha_1^i otimes cdots otimes alpha_k^i) (T) =0, , , 1le i le j }.$$
Counting dimensions, we expect that the map is dominant when
$$j , (n_1 + n_2 cdots +n_k – k) ge j , left( n_1 cdots n_k – j right)$$
$$Rightarrow , , , j ge n_1 cdots n_k – (n_1 + n_2 cdots +n_k – k) .$$
To see that our expectation is correct in the case where equality holds, it is enough to show generically finite fibers. Let
$$L = Phi ( bs{alpha}) = biglangle alpha_1^1 otimes cdots alpha_k^1, , ldots , , alpha_1^j otimes cdots alpha_k^j bigrangle^perp ,$$
so that $$PP (L^perp)$$ is a $$(j-1)$$-secant of the Segre variety
$$Sigma_{n_1-1, ldots, n_k-1}subset PP (V_1 otimes cdots otimes V_k ),$$
which has dimension $$(n_1 + n_2 cdots +n_k – k).$$ Since the dimensions of $$PP (L^perp)$$ and $$Sigma_{n_1-1, ldots, n_k-1}$$ are complementary, the classical "trisecant lemma" implies for generic $$bs{alpha}$$ that
$$PP (L^perp ) cap Sigma _{n_1-1, ldots, n_k-1} = { [alpha_1^1 otimes cdots alpha_k^1], , ldots , , [alpha_1^j otimes cdots alpha_k^j] }.$$

(See eg. Prop 1.3.3 in Russo’s "Geometry of Special Varieties." The statement assumes characteristic $$0,$$ though I’m not sure if it’s necessary for me.)

Thus, if $$Phi (bs{alpha} ) = Phi (bs{alpha}’)$$ for $$bs{alpha}$$ generic, then $$bs{alpha }$$ equals $$bs{alpha}’$$ after permuting some of the $$j$$ factors in the domain.

I discovered the above argument for the case where $$k=2$$ and $$(n_1, n_2) = (3,3),$$ in a context apparently unrelated to tensors, and was surprised that it generalized with little effort. I am thus wondering if it is a well-known fact or if there is some simpler explanation.

A seemingly more difficult question I am also interested in: describe the constructible set $$im Phi .$$