Let $\pi:\mathcal{E}\rightarrow U$ be a family of elliptic curves with level $n$ structure (in the sense of Deligne-Rapoport) where $U\subseteq C$ is some (non-empty) Zariski open set of a smooth complex projective curve $C$. Let us take $\tilde{\mathcal{E}}$ the minimal proper regular model of $\mathcal{E}$ over $\mathbf{C}((t))$ where $t$ is (an appropriate choice) a local parameter around some point on $C$. Embedd $\tilde{\mathcal{E}}$ in $\mathbf{P}^m(\mathbf{C}((t)))$ (for some appropriate $m$) and assume that the defining equations of the embedding are defined over $\mathbf{C}[[t]]$. Now one may reduce the scheme $\tilde{\mathcal{E}}$ modulo $t$ to obtain a scheme $E'$ over $Spec(\mathbf{C})$. Assume that $E'$ is not smooth.

Q1 Is it "possible" (for us humans) to write down explicit equations for such an embedding (at least for small values of $n$)?

Q2 How does one prove (algebraically and/or analytically) that $E'$ is isomorphic (as algebraic variety) to a cyclic configuration of $kn$-copies of $\mathbf{P}^1(\mathbf{C})$ where $k$ is a suitable integer.