TITLE: Deformations of hypersurfaces
QUESTION [8 upvotes]: Suppose I have a smooth hypersurface $X$ in $\mathbb{P}^n$ which is invariant under a (say finite) group $G$ of projective transformations. What can be said about the action of $G$ on the deformation space $H^1(X,T_X)$? I could imagine that many examples (especially with $dim(X)=1$ or $dim(X)=2$) have been worked out, if this is of any interest at all.

REPLY [5 votes]: Suppose that the action of $G$ on the smooth hypersurface $X$ of degree $d$ in $\mathbb P^n$ comes from a linear action of $G$ on $k^{n+1}$, as in Francesco's example. Assume also that $n \geq 3$, and $d \geq 2$. From the conormal sequence
$$
0 \longrightarrow \mathrm T_X \longrightarrow \mathrm T_{\mathbb P^n}{\mid}_X \longrightarrow
\mathcal O_X(d) \longrightarrow 0
$$
we get an exact sequence
$$
\mathrm H^0(X, \mathrm T_{\mathbb P^n}{\mid}_X)\longrightarrow 
\mathrm H^0(X, \mathcal O_X(d)) \longrightarrow \mathrm H^1(X, \mathrm T_X)\ .
$$
One can show that the homomorphism $\mathrm H^0(X, \mathcal O_X(d)) \longrightarrow \mathrm H^1(X, \mathrm T_X)$ is surjective, except in the single case $n = 3$, $d = 4$, where there is a 1-dimensional cokernel. Let us exclude this particular case (which can also be treated).
For each $i \geq 0$, let $V_i$ be the space of forms of degree $i$ in $n+1$ variables; there is a natural action of $G$ on $V_i$. In representation theoretic terms, $V_i = \mathop{\rm Sym}^i(k^{n+1})^\vee$. Let $f \in V_d$ an equation for $X$ and $L$ the substspace generated by $f$. From the Euler sequence
$$
0 \longrightarrow \mathcal O_X \longrightarrow \mathcal O_X(1)^{n+1} \longrightarrow 
\mathrm T_{\mathbb P^n}{\mid}_X \longrightarrow 0
$$
we get a surjection $\mathrm H^0(X, \mathcal O_X(1))^{n+1} \to
\mathrm H^0(X, \mathrm T_{\mathbb P^n}{\mid}_X)$; thus $\mathrm H^1(X, T_X)$ can be interpreted as the cokernel of a map $\phi \colon \mathrm H^0(X, \mathcal O_X(1))^{n+1} \to \mathrm H^0(X, \mathcal O_X(d))$. We have $\mathrm H^0(X, \mathcal O_X(1)) = V_1$ and $\mathrm H^0(X, \mathcal O_X(d)) = V_d/L$; furthermore, by unwinding the definitions one can show that the map $\phi \colon V_1^{n+1} \to V_d/L$ sends $(\ell_0, \dots, \ell_n)$ into the class of $\ell_0f_{x_0} + \cdots + \ell_n f_{x_n}$.
So, one can describe $\mathrm H^1(X, \mathrm T_X)$ as the quotient of $V_d$ modulo the subspace generated by $f$ and by the classes of the form $\ell_0f_{x_0} + \cdots + \ell_n f_{x_n}$, where the $\ell_i$ are homogeneous of degree 1. If the characteristic of $k$ does not divide $d$, then from Euler's formula we see that $f$ is of the form $\ell_0f_{x_0} + \cdots + \ell_n f_{x_n}$, so we don't need to add it.
This gives a description of the action of $G$ on $H^1(X, \mathrm T_X)$, which allows to compute it, at least in simple cases (the calculations could become unwieldy, particularly in the non-abelian case).