Definition
Let \(G\) be a graph decorated by a map \(\mathbf g:V(G)\to \mathbb N\), associating to each vertex its genus. Then there is an algebraic stack \(\mathbf M(G, \mathbf g)\) of \((G, \mathbf g)\)-marked pre-stable curves with obejcts over a \(B\)-scheme \(S\) given by
- for each vertex \(v\in V(G)\), a family of pre-stable curves \(C_v\to S\) of genus \(\mathbf g(v)\), together with marked sections \(x_L:S\to C_v\) defined by the legs \(L \in L(G)\) with \(v \in L\)
- for each edge \(E \in E(G)\) with vertices \(v,w\), a pair of marked sections \(y_v,y_w\) of \(C_v\to S\), \(C_w\to S\), respectively.
Basically, it’s the stack parameterizing marked curves whose dual graphs are \(G\) with the prescribed genera.