TITLE: Automorphisms and infinitesimal deformations of a smooth complete intersection
QUESTION [5 upvotes]: Let $X\subset\mathbb{P}^{n+c}$ be a smooth complete intersection of dimension $n$. 
Is it known when $Aut(X)$ is finite ?
Does there exist a formula for the dimension of the tangent space to the space of first order infinitesimal deformations $H^1(X,T_X)$ ?
I am especially interested in the case of a codimension two complete intersection of two quadrics.

REPLY [4 votes]: I can suggest you a shorthand for your second question, in case you need a practical answer.
First, if $X$ is a smooth complete intersection, let me denote by $A_X$ the affine cone over it, by $U_X= A_X \smallsetminus v$ the punctured affine cone and $\pi: U_X \to X$ the natural projection. We have then a standard short exact sequence (see for example Schlessinger's paper "On Rigid Singularities") $$ 0 \to \mathcal{O}_{U_X}\to T_{U_X} \to \pi^* T_X \to 0 $$
since we have a trivialization of the relative tangent by the Euler vector field.
Now, for a smooth complete intersection of dimension $\geq 2$ we have $$H^1(U_X, T_X) \cong T^1_{A_X}$$ where the object on the right hand side is the (graded) module of first order deformations of the affine cone $A_X$. Now, passing in cohomology and since  in general $$H^q(U_X, \pi^* \mathcal F)= \bigoplus_{k \in \mathbb Z} H^q(X, \mathcal F(k))$$ we have (when we restrict to the degree 0) $$ \cdots \to H^1(X, \mathcal{O}_X) \to (T^1_{A_X})_0 \to H^1(X, T_X) \to H^2(X, \mathcal O_X) \to \cdots $$
For a complete intersection of dimension $>2$ $H^i(X, \mathcal O_X)=0$, $i=1,2$, and thus we have $$(T^1_{A_X})_0 \cong H^1(X, T_X).$$
Now, the point is that the dimension of the graded component of the $T^1_{A_X}$ are super-easily computable using a bit of computer algebra: for example SINGULAR is extremely efficient in doing this (see for example https://www.singular.uni-kl.de/Manual/latest/sing_874.htm#SEC925), and basically you can get your answer with a couple of 'one-line command'.