$D_{J;u}^{NY}$ whenever $([u],w)\!\in\!\ov\M_{g,\f}(X,\be;J)$. For example, this is the case if $X$ is the blowup of $\P^n$, with $n\!\ge\!2$, at a point, $Y\!\approx\!\P^{n-1}$ is the exceptional divisor, $S\!=\!\{1,2\}$, $g\!=\!0$, $\be_Y\!\in\!H_2(Y;\Z)$ is the homology class of a line in the exceptional divisor, $\be\!=\!\io_{Y*}(\be_Y)$, $a_1,a_2\!=\!0$, and $f_1,f_2\!:\P^1\!\lra\!X$ are parametrizations of proper transforms of two distinct lines in $\P^n$ passing through the center of the blowup. In this particular case, $$\ov\M_{0,\f}(X,\be;J)=\ov\M_{0,\f}(Y,\be_Y;J)$$ and $\cok(D_J^{NY})$ is the zero vector bundle. Thus, if $\bar\ell$ denotes the homology class of $f_1$ and $f_2$, $$\big(\bar\ell,\bar\ell\big)_{0,\be}^X=
\big((\tau_{a_j}\ka_j)_{j\in S}\big)_{0,\be}^X
=\bC_{0,\f}(Y,\be_Y)
=\,^{\pm}\big|\ov\M_{g,\f}(Y,\be_Y;J)\big|
=\big(\bar\ell\cap Y,\bar\ell\cap Y\big)_{0,\be_Y}^Y
=1;$$ see Footnote \[inY\_ft\].

Various special cases of Theorem \[main\_thm\], such as those in Examples \[CY\_eg\]-\[transversetoY\_eg\], are standard in the algebraic setting and are used in [@BP], [@KlP], and [@P1], for example. Some special cases of Theorem \[main\_thm\] have appeared in the symplectic setting as well, including in [@LiZ], [@McTo], and [@Taubes]. Examples \[inY\_eg\] and \[transversetoY\_eg\] generalize Example \[CY\_eg\] in two opposite directions. Corollary \[LeP\_crl\] below, which applies this theorem in the setting of [@LeP], is yet another special case of Example \[transversetoY\_eg\]. The full statement of Theorem \[main\_thm\] mixes the two extreme cases of Examples \[inY\_eg\] and \[transversetoY\_eg\].\
The striking conclusion of [@LeP] is that all 