Has anyone come across anything along the following lines?

Let $X_1(p)$ be the compactification of the quotient of upper half plane by $\Gamma_1(p)$ for some unspecified large prime. Let $X_1(p) \to \mathbb{P}^n$ be some embedding into some projective space. Has anyone explicitly constructed a flat family of dimension one subschemes of $\mathbb{P}^n$ such that generic fiber is isomorphic to $X$ but a special fiber is something degenerate, yet explicit. For example, has anyone calculated explicitly Grobner basis or some Okounkov-Newton style binomial degeneration?

My interest is primarily in $\Gamma_1(p)$ and embedding by weight one Eisenstein series, but other embeddings and other modular groups (like $\Gamma(p)$ and $\Gamma_0(p)$) are interesting as well.