I am reading Illusie's lecture notes "topics in algebraic geometry", and I have difficulty in following his proof of unobstructedness of deformation of curves. Here is the statement of the proposition:

**Prop** Let $f_0: X_0 \to Y_0$ be a smooth proper morphism with relative dimension 1, and $i : Y_0 \to Y$ a first-order thickening with ideal $I$. If moreover $Y$ is affine, then there always exists a lifting of $X_0$ over $Y$ .

**Proof**. First, since $Y_0$ is affine, we note that
$H^q(X_0, T_{X_0/Y_0}
\otimes f_{0}^{*}I)= \Gamma (Y_0,R^qf_{0*}(T_{X_0/Y_0}\otimes f_{0}^{*}I))$

By Zariski’s main theorem, for any $q > 1$,

$R^qf_{0*}(T_{X_0/Y_0}\otimes f_{0}^{*}I))=0$ (*)

Hence the obstruction $o(f_0, i)\in H^2(X_0, T_{X_0/Y_0}\otimes.
f_0^*I)$ vanishes. #

And I wonder how to derive (*) by Zariski's main theorem.

Thank you!