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 morphism 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 $\mathcal{E}xt^2([W_1 \to \Omega_{\mathcal{X}/S}] , \mathcal{O}_X)$?

Thank you for your answers.



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