Stable n-pointed algebraic curves and the moduli functor

Let $S$ be a scheme and let $g,n\ge 0$ satisfy $2g-2+n>0$. A nodal curve family of genus $g$ over $S$ is a proper, flat, finitely presented morphism $f:C\to S$ of relative dimension one whose geometric fibres are reduced, connected, one-dimensional curves with only ordinary nodes and arithmetic genus $g$, where arithmetic genus means $1-\chi (C_s,{\mathcal{O}}_{C_s})$. A marking is a section $p_i:S\to C$; the markings are ordered, pairwise disjoint, and factor through the relative smooth locus.

A stable $n$-pointed curve family is such data $(f:C\to S,p_1,\dots ,p_n)$ for which the relative dualizing sheaf with markings ${\omega }_{C\mathrm{/}S}(p_1+\cdots +p_n)$ is relatively ample. An isomorphism over $S$ is an $S$-isomorphism of the curve families carrying each $p_i$ to the corresponding marking. The moduli functor ${\bar{\mathcal{M}}}_{g,n}$ assigns to $S$ the groupoid of these families and isomorphisms; this definition does not assert the functor is an algebraic stack or is proper.

For a geometric fibre, let $E$ be an irreducible component, let $g(E)$ be the genus of its normalization, and let $s(E)$ count its markings and the branches of nodes on that normalization. The stability criterion is $2g(E)-2+s(E)>0$ for every $E$: rational components have at least three special points, genus-one components at least one, and components of genus at least two need no special point. Indeed, the degree of the restricted dualizing line bundle is $2g(E)-2+s(E)$, and ampleness on a proper nodal curve is equivalent to positive degree on every irreducible component.

The stability range excludes the unstable constant-curve cases $(g,n)=(0,0),(0,1),(0,2),(1,0)$. For $(g,n)=(0,3)$ the fibre is a three-pointed projective line up to unique isomorphism; for $(g,n)=(1,1)$ the single marking removes translations. These are definition-level boundary cases, not properness claims.