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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 for small values of $n$ (larger than $3$ and $4$ since these cases have been worked out by Igusa)?

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.

first addendum

This a rather a long comment to question 1. So the universal elliptic curve with full level 3 structure (in rings where $3$ is invertible) can be written as $$ X^3+Y^3+Z^3-3tXYZ=0 $$ where $t\in \mathbf{P}^1(\mathbf{C})-\{1,\zeta,\zeta^2\}$ where $\zeta=e^{2\pi i/3}$. For example at $t=0$ the family "degenerates" to the smooth cubic $$ X^3+Y^3+Z^3=0. $$ When $t=1$, the family degenerates to the singular cubic $$ X^3+Y^3+Z^3-3XYZ=0. $$ The singular points are located at $(\zeta,\zeta^2,1)$, $(\zeta^2,\zeta,1)$ and $(1,1,1)$. In fact one has the factorization (unless I made some mistake) $$ (x+y+z)(x+\zeta y+\zeta^2 z)(x+\zeta^2 y+\zeta z)=X^3+Y^3+Z^3-3XYZ, $$ and one readily sees that we get a cyclic configuration of $3$ copies of $\mathbf{P}^1(\mathbf{C})$.

So in general, it would be nice to know why, systematically, a degenerate fiber of a family of elliptic curves (with level $n$ structure) is a cyclic configuration of $n$-copies of $\mathbf{P}^1(\mathbf{C})$. Is there a purely algebraic proof of this result?

second addendum

Of course topologically, the "most natural" degeneration of a complex torus $T$ with level $n$-structure should consist in pinching $n$-distinct points on $T$, so that we get a "necklace" of consisting of $n$ 2-spheres. But I don't quite see (algebraically and coomplex analytically) why the level $n$ structure forces the degeneration to be a necklace.

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What is it that you want that the Tate curve does not provide? – Felipe Voloch Sep 11 '13 at 1:55
Suppose that you are an Italian geometer who does not know about the Tate curve. How do you see that a family of elliptic curves with full level $n$ structure degenerates to a cyclic configuration to $kn$ copies of $P^1$? – Hugo Chapdelaine Sep 11 '13 at 10:06
I'm not sure if this is relevant, but for $n=5$, there is a nice expression for these level $5$ structures using Pfaffians, which is discussed in many places, for example Fisher's paper or (to self-advertise) my paper with Gruson (section 2.2) The latter has an explicit family over $P^1$ whose singular fibers are pentagons – Steven Sam Sep 12 '13 at 2:06
Thanks a lot Steven for these two references! – Hugo Chapdelaine Sep 12 '13 at 11:37

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