# Computing a Gromov-Witten invariant

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 $$q$$).