half-edges correspond to the markings. The closure of the locus of curves with a given dual graph in is isomorphic to the stack quotient of a product of compactified moduli spaces of curves by a finite group. In the product the factor corresponding to a vertex v has genus gv taken from the labelling and number of markings equal to the number of outgoing edges and half-edges at v. The total genus g is the sum of the gv plus the number of closed cycles in the graph. Stable curves whose dual graph contains a vertex labelled by (hence all other vertices have and the graph is a tree) are called "rational tail" and their moduli space is denoted . Stable curves whose dual graph is a tree are called "compact type" (because the Jacobian is compact) and their moduli space is denoted . See also Moduli of marked curves Witten conjecture Tautological ring Grothendieck–Riemann–Roch theorem References Classic references Books on moduli of curves Cohomology and intersection theory External links "Topology and geometry of the moduli space of curves" "Moduli of Stable Maps, Gromov-Witten Invariants, and Quantum Cohomology" Moduli theory Algebraic varieties