  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.
