Compactification of $M_{1, 1}$

Question

Let $M_{1, 1, \mathbb{Z}}$ be a moduli stack of elliptic curves. Denote $M_{1, 1, k}$ its base change to a field $k$. We have the map $j: M_{1, 1, k} \to \mathbb{A}^1_{k}$ which realizes the affine line as a coarse moduli space (via the $j$-invariant). I'm interested in the compactification $\overline{M}_{1, 1, k}$. How to describe it explicitly? It is known that its coarse moduli space is $\mathbb{P}^1_{k}$. What is the fiber at $\infty$? Any references are greatly appreciated!

Answer 1

The Deligne-Mumford compactification is by nodal curves, so the fiber at $\infty$ is a nodal curve of genus $1$ with $1$ marked point $p$, which is unique up to isomorphism since its normalization is a copy of $\mathbb P_k^1$ with $3$ marked points ($p$, and the two points lying over the node), and the marked points can always be taken as $0,1,\infty$ after applying an automorphism.

Note 1: just to be clear, we do have $\mathbb P_k^1$ appearing both as the moduli space AND as (the normalization of) a fiber of the universal curve over it. I only mention this for the same reason one sometimes distinguishes between $4$ arising as $2+2$ and $4$ arising as $2^2$: it's one of those things that, while baked into the nature of reality, seems designed to cause confusion.

Note 2: I'm by far best informed about the case where $k$ is algebraically closed, but I don't think there are subtleties here. The only potential issue I can think of is that genus $0$ does not necessarily imply $\cong \mathbb P^1$; but such curves have no $k$-points, and you have assumed the existence of the $k$-point which is marked in the initial setup.