# The proof of unobstructedness of deformations for curves

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!

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Proposition. Let $f \colon X \to Y$ be a projective morphism of noetherian schemes, and let $r= \textrm{max} \{\dim X_y | y \in Y \}$. Then $R^qf_* (\mathscr{F})=0$ for all $q >r$ and for all coherent sheaves $\mathscr{F}$ on $X$.
Now by projection formula we have $$R^qf_{0*}(T_{X_0/Y_0}\otimes f_{0}^{*}I))=R^qf_{0*}(T_{X_0/Y_0}) \otimes I,$$ so you can apply the previous proposition with $r=1$ and $\mathscr{F}=T_{X_0/Y_0}$ in order to get the desired vanishing.