Moduli Space of Stable Curves

The moduli space of stable curves parametrizes stable marked curves of a fixed arithmetic genus with labeled markings, where . The notation refers to the moduli stack, which retains automorphisms and families. Its coarse moduli space records isomorphism classes. The open locus parametrizes smooth curves, and the boundary adds nodal stable curves.

Its dimension is . A stable graph records the genera of irreducible components, their nodes as edges, and the marked points as legs. Such graphs index boundary strata. The tautological ring is generated using cotangent classes and the operations of forgetting markings and gluing nodes. The Pixton completeness conjecture concerns relations among these classes in rational Chow.