Definition
An \(n\)-pointed stable map consists of a connected marked curve \((C,p_1,...,p_n)\) and a morphism \(f:C\to X\) satisfying the following properties.
- The only singularities of \(C\) are ordinary double points
- The \(p_i\) are all diinct ordered smooth point of \(C\) (so an isomorphism \(C\cong C'\) must map \(p_i \mapsto p_i'\))
- If \(C_i\) is a component of \(C\) such that \(C_i\cong \mathbb P^1\) and \(f\) is constant on \(C_{i}\), then \(C_i\) contains at least \(3\) special (nodes or marks) points.
- If \(C\) has arithmetic genus \(1\) and \(n=0\) then \(f\) is not constant.
Note that given the first and second axioms above, the third and fourth together are equivalent to requiring that \(C\) has finitely many automorphisms.