Hadamard Products and Varieties Which Are Strongly Concise for All Systems of Coordinates

We fix the Hadamard product (coordinate-wise multiplication) of a projective space. The strongly concise embedded varieties are the embedded varieties X such that all the points of the projective space have finite Hadamard X-rank. We prove that for all $n\ge 2m+1$, every m-dimensional projective manifold Y may be embedded in a projective space of dimension n as a strongly concise variety for all changes of coordinates. We prove that Y may be embedded in such a way that for a certain coordinate system not only it is not strongly concise, but the general point of the projective space has infinite Hadamard X-rank. We use previous works by D. Antolini, G. Montúfar, and A. Oneto.