The main result of the paper is the following. Let \(X\) be a \(k\)-dimensional complex space embeddable into \(\mathbb{C}^{n}\) and \( 0 < k < n-1\). The authors construct holomorphic families of proper holomorphic embeddings of \(X\) into \(\mathbb{C}^n\) parametrized by \(\mathbb{C}^{n-k-1},\) such that for any two distinct parameters \(w_1,w_2 \in \mathbb{C}^{n-k-1}\) the corresponding embeddings \(X \hookrightarrow \mathbb{C}^{n}\) are not equivalent.
The main idea of the proof is to start with the constant family of embeddings and use a parametric version of the standard way of producing nonstraightenable holomorphic embedding of \(\mathbb{C}^{k} \) into \(\mathbb{C}^{n}\) by using a sequence of holomorphic automorphisms. As in the nonparametric setting the embeddings contain certain countable unions of points on the spheres that are used to prove growth conditions on a holomorphic automorphism that maps one embedding from the family to another. At this point an interpolation result is needed and the core problem in proving it is to bring parametrized points into the standard position. The main tools that are used in the proof are the Andersén-Lempert theory and the Oka-Grauert-Gromov homotopy theory.