817_10-11

Catalogue

1. 1. Tangent space of $\mathrm{Mor}(Y,X)$
2. 2. dimension of tangent space
3. 3. $\mathrm{Mod} p$ reduction

Tangent space of $\mathrm{Mor}(Y,X)$

Goal: proof that

$$ T_{Mor(Y,x),[f]} \cong H^0(Y,\mathcal{H}om(f^*\Omega _X,\mathcal{O}_Y) $$

dimension of tangent space

$$ \dim T_{Mor(Y,x),[f]} \geq h^0(Y,f^T_X) - h^1(Y,f^T_X) $$

Inparticular, if $h^1(Y,f^*T_X)=0$, then smooth at $[f]$

skip this proof.

$\mathrm{Mod} p$ reduction