Some background that is not necessary for answering the question:
Let $X = \mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ be a threefold. This is a $\mathbb{P}^1$-bundle over $\mathbb{P}^2$. Let $f$ be the cohomology class of the fiber. This bundle has a section whose image has normal bundle $\mathcal{O}(-2)$ as a hypersurface in $X$. Let $\beta \in H_2(X, \mathbb{Z})$ be the class of a line on this hypersurface. A localization computation gives me the Gromov-Witten invariant $GW^X_\beta\langle f \rangle = -1$. I would like to directly compute this invariant using obstruction theory.
My actual question:
The moduli space of lines in the hypersurface in question (which is isomorphic to $\mathbb{P}^2$) is the dual $(\mathbb{P}^2)^*$. The obstruction bundle on this dual space is the bundle whose fiber over each point (which is a line $i:L \hookrightarrow \mathbb{P}^2$) is $H^1(L, i^*\mathcal{O}(-2))$. How do I see directly that this bundle is in fact $\mathcal{O}(-1)?$
The universal line (call it $Z$) is a divisor of type $(1,1)$ in the product $\mathbb{P}^2 \times (\mathbb{P}^2)^*$. Consequently, there is an exact sequence (the Koszul complex) $$ 0 \to \mathcal{O}_{\mathbb{P}^2 \times (\mathbb{P}^2)^*}(-1,-1) \to \mathcal{O}_{\mathbb{P}^2 \times (\mathbb{P}^2)^*} \to \mathcal{O}_{Z} \to 0. $$ You are asking about the computation of $R^1q_*(p^*\mathcal{O}(-2))$, where $p$ and $q$ are the projections of $Z$. Using the above resolution and the projection formula gives $$ R^1q_*(p^*\mathcal{O}(-2)) = R^2q_*(\mathcal{O}_{\mathbb{P}^2 \times (\mathbb{P}^2)^*}(-3,-1)) = H^2(\mathbb{P}^2,\mathcal{O}(-3)) \otimes \mathcal{O}_{(\mathbb{P}^2)^*}(-1) = \mathcal{O}_{(\mathbb{P}^2)^*}(-1) $$ (abusively, the projections of $\mathbb{P}^2 \times (\mathbb{P}^2)^*$ to the factors are also denoted by $p$ and $q$).