some}\quad
d_1,\ldots,d_k\in\Z^+.$$ By \_ref[emptyspace\_e]{}, $\lr{c_1(TX),\be_i}\!\ge\!0$ for all $i\!=\!1,\ldots,k$. Thus, $$\sum_{i=1}^{i=k}\lr{c_1(TX),\be_i}\le \lr{c_1(TX),\be}.$$ The dimension-counting argument of [@McS Section 6.6] then shows that $k\!=\!1$ and $d_1\!=\!1$. It then follows that the image of $u$ is an irreducible $J$-holomorphic curve of degree $\be$ and genus $h\!<\!g$ that meets each of the maps $f_j$ with $j\!\in\!S$.\
While degree $\be$ genus $h\!<\!g$ $J$-holomorphic curves meeting the maps $f_j$ can certainly exist for a generic $J$, they cannot be limits of other degree $\be$ curves meeting the maps $f_j$ by the $\nu_r\!=\!0$ case of Proposition \[horreg\_prp\] for the following reason. If $$\big([u],(w_j)_{j\in S}\big)\in\ov\M_{g,\f}(X,\be;J)
-\M_{g,\f}^*(X,\be;J),$$ the domain of $u$ consists of two or more irreducible components. Furthermore, by the previous paragraph, the restriction of $u$ to all components, except for one, is constant; let $u_{\eff}$ denote the of $u$, i.e. the non-constant restriction. The domain $\Si_{u_{\eff}}$ of $u_{\eff}$ is a smooth curve of genus $h\!<\!g$ with distinct points $(z_j(u_{\eff}))_{j\in S}$ that are mapped to $(\ev_j(u))_{j\in S}$ by $u_{\eff}$. Thus, $$\big([u_{\eff}],(w_j)_{j\in S}\big)\in\M_{h,\f}^*(X,\be;J);$$ by the previous paragraph, $u_{\eff}$ is an embedding onto a smooth $J$-holomorphic curve $Y$ of genus $h$ degree $\be$ meeting the maps $f_j$. This implies that removing a node from $\Si_{u_{\eff}}$ disconnects $\Si_u$.[^7] Since the total 