(Stable map)
Let \(X\) be a projective variety over \(\CC\) and \(\beta \in H_2(X, \ZZ)\). A stable map of genus \(g\) with \(n\) marked points and class \(\beta\) is a tuple \((C; p_1, \ldots, p_n; f)\) where \(C\) is a connected projective curve of arithmetic genus \(g\) with at worst nodes, \(p_1, \ldots, p_n\) are distinct smooth points of \(C\), and \(f \colon C \to X\) is a morphism with \(f_*[C] = \beta\), such that the automorphism group of the tuple is finite. Equivalently, every irreducible component of \(C\) of genus \(0\) contracted by \(f\) contains at least three special points (nodes or marked points), and every contracted component of genus \(1\) contains at least one.
(Kontsevich)
Stable maps are parametrized by a proper Deligne–Mumford stack \(\overline{\mathcal{M}}_{g,n}(X, \beta)\). For \(X\) a point it is the moduli stack \(\overline{\mathcal{M}}_{g,n}\) of stable marked curves.
The Gromov–Witten invariants of \(X\) are integrals over the virtual fundamental class of \(\overline{\mathcal{M}}_{g,n}(X, \beta)\) of pullbacks of cohomology classes along the evaluation maps \(f \mapsto f(p_i)\). For \(X = \PP^2\), \(g = 0\) and \(\beta = d[\text{line}]\) with \(n = 3d - 1\) point classes, the invariant is the number \(N_d\) of rational plane curves of degree \(d\) through \(3d - 1\) general points.