Definition
(Class). The data \(\beta = (g, A, n, u_p)\) is called the class associated to a log map \((C, f, p_1,...,p_n)\) where
- \(g \) is the genus of the source curve
- \(A \) is the curve class \(f_*[C]\)
- \(n\) is the number of marked points
- \(u_{p_i}\) is the contact order at the marked point \(p_i\)
The moduli space \(M(X/B, \beta)\) is then the moduli space of stable maps of class \(\beta\). The data \(\underline \beta\) is precisely the combinatorial data specified in ordinary Gromov-Witten theory. See definitions 2.15-2.16 in the Decompositions paper. It is this moduli space whose degenerate fiber is decomposed in the main decomposition theorem.