Definition
Let X be a projective algebraic variety ad let \(S\) be a scheme over \(\mathbb C\). An \(n\)-pointed stablemap over \(S\) is a flat proper morphism \(C\to S\) together with \(n\) sections \(s_1,...,s_n\) an a map \(f:C\to X\) suh that for each geometric point \(s\) of \(S\), the restriction \(f_s:C_s\to X\) of \(f\) to the geometric fibers of \(C\) over \(s\), together with the images of the sections \(s_i\), defines a stable map. Furthermore:
- We say that \(f:C\to X\) has genus \(g\) if for each geometric point \(s\) of \(S\), the curve \(C_s\) has arithmetic genus \(g\).
- Given a homology class \(\beta\in H_2(X,\mathbb Z)\), we say that \(f:C\to X\) has class \(\beta\) if for each geometric point \(s\) of \(S\), \((f_s)_*[C_s] = \beta\).