Injection dimensions of projective varieties

We explore injective morphisms from complex projective varieties $X$ to projective spaces $\mathbb{P}^s$ of small dimension. Based on connectedness theorems, we prove that the ambient dimension $s$ needs to be at least $2 \dim X$ for all injections given by a linear subsystem of a strict power of a line bundle. Using this, we give an example where the smallest ambient dimension cannot be attained from any embedding $X \hookrightarrow \mathbb{P}^n$ by linear projections. Our focus then lies on $X = \mathbb{P}^{n_1} \times \ldots \times \mathbb{P}^{n_r}$, in which case there is a close connection to secant loci of Segre--Veronese varieties and the rank $2$ geometry of partially symmetric tensors, as well as on $X = \mathbb{P}(q_0,\ldots,q_n)$, which is linked to separating invariants for representations of finite cyclic groups. We showcase three techniques for constructing injections $X \to \mathbb{P}^{2\dim X}$ in specific cases.