Proper Deligne–Mumford stack of stable pointed curves

The families in [F1] have the source's stable pointed-curve convention: on a normalized component of genus $h$ with $s$ special points the marked dualizing degree is $2h-2+s>0$. In the unpointed range $g\ge 2$ this excludes precisely rational tails and bridges, agreeing with the stable-stack convention of [F2]. Thus [F2] gives the proper smooth Deligne–Mumford base for $g\ge 2,n=0$. For $(g,n)=(0,3)$, a geometric stable fibre with $v$ rational components has a tree as its dual graph and $3+2(v-1)=2v+1$ special points, whereas stability requires at least $3v$; hence $v=1$. The family-level identification ${\bar{\mathcal{M}}}_{0,3}=\mathrm{Spec}\mathbb{Z}$ is the base identification in [A1], not an inference from geometric points alone. These are the non-genus-one bases.

For fixed $g\ge 2$, use the approved contraction/stabilization identification in [A1]: ${\bar{\mathcal{M}}}_{g,n+1}\to {\bar{\mathcal{M}}}_{g,n}$ is the representable proper universal stable curve. Properness follows by composition and induction from the unpointed base in step 1.1. The same construction gives ${\bar{\mathcal{M}}}_{0,n+1}$ from ${\bar{\mathcal{M}}}_{0,n}$ for $n\ge 3$, starting at its $(0,3)$ base. The construction includes the rational tail or bridge inserted when the new mark meets an old mark or a node, so no geometric case is omitted.

For smoothness and the boundary induction, start with the unpointed boundary normal-crossings assertion explicitly imported in [A1], and with the empty boundary of ${\bar{\mathcal{M}}}_{0,3}$. On a smooth base chart with nodal-boundary parameters $t_1,\dots ,t_k$, the universal curve is smooth near each old section, while near a node the source chart is $xy=t_i$. Eliminating the independent parameter $t_i$ gives a smooth total stack over $\mathrm{Spec}\mathbb{Z}$. Representability of the universal curve also preserves the Deligne–Mumford property: pull an étale scheme atlas back from its Deligne–Mumford base, obtaining an étale scheme atlas of the total stack. This does not use finite automorphism groups as a substitute for unramifiedness.

The boundary in the new total stack is the union of the inverse image of the old boundary and the images of the old sections, exactly as in [A1]. Near an old section choose its smooth fibre coordinate $u$; the union has equation $u{t}_1\cdots t_k=0$. Near a node the old sections are absent, and $xy=t_i$ changes the pulled-back boundary equation to $xy{\prod }_{j\mathrm{\ne }i}{t}_j=0$. These are relative normal-crossings equations. Hence both smoothness and the stated boundary property propagate in the two inductions of step 2.1. The starting unpointed boundary property here comes from the full approved Knudsen theorem.

For the genus-one base use the precise argument imported in the proof of [A1]. Its fixed three-pointed genus-one curve $E$ has three rational components and no automorphism fixing its distinguished points. Clutching it to the mark of a varying $(1,1)$-curve gives a closed immersion ${\bar{\mathcal{M}}}_{1,1}\hookrightarrow {\bar{\mathcal{M}}}_{2,2}$. Its image is locally the intersection of four nodal-boundary branches, and its own nodal locus is the intersection with a fifth branch. The ambient stack is proper, smooth and Deligne–Mumford with relative normal-crossings boundary by steps 2.1–4.1. Thus this source stratum is proper, smooth and Deligne–Mumford, with its stated relative nodal divisor. This uses the approved clutching argument within Theorem 2.7's proof; it is not a new independently assumed clutching theorem.

Starting from this $(1,1)$ base, the universal-curve induction gives every $(1,n)$ with $n\ge 2$, and the same local calculations in steps 3.1–4.1 preserve smoothness, Deligne–Mumford structure and relative normal crossings. Together with the genus-zero and $g\ge 2$ inductions these are all $2g-2+n>0$ cases. Properness in every case is algebraic properness over $\mathrm{Spec}\mathbb{Z}$. The construction asserts no analytic compactness comparison.