= Rank bound for a nonsingular complex bilinear map
{title2=$\operatorname{rank}\phi\geq\dim U+\dim V-1$}
Let $U,V$ be nonzero finite-dimensional complex <vector spaces>, and let $\phi:U\otimes V\to W$ have injective restrictions when either nonzero factor is fixed. It induces a map from $\mathbb{CP}^{a-1}\times\mathbb{CP}^{b-1}$ to $\mathbb{CP}^{r-1}$, where $a=\dim U$, $b=\dim V$, and $r=\operatorname{rank}\phi$. Pullback of the degree-two generator is $x+y$, since both factor restrictions are projective linear embeddings. Its $(a+b-2)$th power is $\binom{a+b-2}{a-1}x^{a-1}y^{b-1}\ne0$ in integral <cohomology>. The target must therefore have complex dimension at least $a+b-2$. Positivity of both factor dimensions is necessary; a zero factor can make the claimed inequality false.
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