Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost complex structure $J$ is said to be $\textit{regular}$ for some $J$-holomorphic curve $u$ (that represents $\beta$) if the linearization of the $\overline{\partial}$ operator is surjective (at $u$).

$\textbf{Question}:$ Is the standard almost complex structure on a del-Pezzo surface a regular almost complex structure?

$\textbf{Remark}:$ One needs the complex structure to be regular in order to conclude that the moduli space of non multiply covered holomorphic maps has the right dimension (and to compute the Gromov Witten invariants). Since the genus zero Gromov-Witten invariants of del-Pezzo surfaces are well known (see the paper by Kontsevic-Mannin http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf and the paper by Pnadharipande-Gottsche http://arxiv.org/abs/alg-geom/9611012), I would imagine this is a well known fact.

$\textbf{Added Later:}$ Suppose $u:\mathbb{P}^1 \longrightarrow X$ is a degree $\beta$ map. Consider $u^* TX \longrightarrow \mathbb{P}^1 $. By Grothendick's theorem this splits holomorphically as a sum of two line bundles $L_1 \oplus L_2 \longrightarrow \mathbb{P}^1$. If one can show that $c_1(L_1) >= -1$ and $c_1(L_2) >= -1$ then the linearization of $\overline {\partial} $ at $u$ is surjective (this is Lemma 3.3.1 in McDuff and Salamon).

Is there an immediate way to see why the two Chern numers are greater than or equal to $-1$?