Curve-counting with fixed source Suppose I fix a smooth projective curve $C$ of positive genus, and I have a smooth projective variety $X$. Do standard tools from GW theory (or any curve-counting theory for that matter) allow me to compute the (virtual) number of maps $C\to X$ with incidence conditions? I have only ever seen GW invariants defined to be zero when the expected homological dimension is positive. But do the usual techniques (e.g. localization if $X$ is toric) allow me to do something like take the forgetful map $\overline{M_{g,n}}(X,\beta)\to \overline{M_{g}}$ and intersect a positive (virtual-) dimensional locus with a fiber and hope to get a number out?
Apologies if this is a very naive question; I am not an expert in curve-counting by any means.
 A: You can compute the GW invariants associated to maps with fixed domain curve $C$ of genus $g$ in terms of genus 0 invariants. The conceptual idea is that the invariants are independent of the choice of the fixed curve $C$, and so one can choose $C$ to be a rational curve with $g$ nodes, and then use the usual gluing axioms to rewrite the invariant as a genus 0 invariant with $2g$ additional insertions. 
Let $X$ be the target, let $C$ be a fixed curve of genus $g$, $\beta\in H_2(X)$, and let $\gamma_i\in H^*(X)$. Then the fixed domain Gromov-Witten invariant would be defined as
$$\langle \gamma_1,\ldots,\gamma_n\rangle_{\beta,C}^X = \int _{[\overline{M}_{g,n} \, (X,\, \beta)]^{vir}} ev_1^*(\gamma_1)\cdots ev_n^* (\gamma_n) \cdot ct^*([C]^\vee)$$
where $ct: \overline{M}_{g,n} \, (X,\, \beta)\to \overline{M}_g$ takes a stable map to its domain curve, with marked points forgotten and unstable components contracted. Here $[C]^\vee \in H^*(\overline{M}_g)$ is the class of the point associated to $C$. Then we can rewrite this in terms of usual genus 0 invariants by the following:
$$
\langle \gamma_1,\ldots,\gamma_n\rangle_{\beta,C}^X = \frac{1}{g!2^g}\sum_{k_1,\ldots,k_g}  \langle \eta_{k_1},\eta^{k_1},\ldots,\eta_{k_g},\eta^{k_g},\,\gamma_1,\ldots,\gamma_n\rangle_{\beta,0}^X 
$$
where  $\eta_k$ is a basis for $H^*(X)$ and $\eta^k$ is the dual basis so that in particular the class of the diagonal in $H^*(X\times X)$ is given by $\sum_k \eta_k \otimes \eta^k$. 
My combinatorial factor out front might not be right. Your mileage may vary. 
