# Questions about the details in the construction of virtual fundamental class

Let $\pi :D \subset \mathcal{X} \to S$ be a flat family of stable curves of genus $g$ with marked points $D$. Let $\mathcal{X} \to X$ be a flat family of stable morphisms in the sense of Kontsevich over $\mathcal{X} \to S$. In Li and Tian's article, they showed at the beginning of section $4$ that the tangent-obstruction complex they constructed is perfect by choosing a sufficiently ample line bundle $\mathcal{L}$ on $X$ then using the following exact sequence: $$0 \to W_2 \to W_1 \to f^*\Omega_X \to 0$$ where $$W_1 = \pi^\*\pi_*\left( \omega_{\mathcal{X}/S}(D)^{\otimes 5} \otimes f^\*(\mathcal{L} \otimes \Omega_X) \right) \otimes \left(\omega_{\mathcal{X}/S}(D)^{\otimes 5}\otimes f^*\mathcal{L}\right)^{-1}.$$

Here I have several questions:

1. Why is $W_2$ locally free?
2. What are reasons they consider $W_1$ as above? For example, why do we consider $\otimes 5$, $\pi^*\pi_\*$...etc.
3. Why do we have the vanishing of $\mathcal{E}xt^2([W_1 \to \Omega_{\mathcal{X}/S}] , \mathcal{O}_X)$?

Any answer has to involve two ingredients: 1) properties of some cohomology groups on a nodal curve C (i.e., the restriction of the sequence you use to a fiber of $\pi$); 2) the theorem of cohomology and base change, e.g. as in Hartshorne section III.12. Do you need an answer on 1), 2) or both? –  Barbara May 7 '13 at 15:06