A stable curve over 
is a connected projective curve with only ordinary nodes and finite automorphism group. In the marked version, the labeled
marked points are distinct and smooth, and automorphisms must preserve them. A rational
irreducible component must have at least three special points, and a genus-one component
must have at least one. Special points count marked points and the branches of nodes
on the normalization. The numerical stability condition for genus 
and 
markings is 
.
Stable curves allow nodal limits of smooth curves while controlling their automorphisms. Their moduli spaces carry tautological classes obtained from marked-point cotangent lines and boundary gluing maps.
