In the following article : "H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964) 347-361", it is shown that in finite characteristic, automorphism groups of smooth hypersurfaces of $\mathbb{P}^N$ are finite (with known exceptions such as quadrics, elliptic curves, K3 surfaces). However, the question of their reducedness is left open. Does anyone know something about it ?


In fact, we need to show that $H^0(X,T\_X)=0$. In characteristic $0$, you can use Bott's theorem to do that. What can you do in finite characteristic ?



Are automorphism groups of hypersurfaces reduced ?
If $X$ is a smooth hypersurface in $\mathbf{P}^{n+1}$ of degree $d$, where $n \ge 1$, $d \ge 3$, and $(n,d) \ne (1,3)$, then $H^0(X,T\_X)=0$ by Theorem 11.5.2 in Katz and Sarnak, *Random matrices, Frobenius eigenvalues, and monodromy*, AMS Colloquium Publications, vol. 45, 1999. So the connected component of the identity of the automorphism group scheme is trivial in these cases. See also Theorem 11.1 in <http://www-math.mit.edu/~poonen/papers/projaut.pdf> ,
which is Poonen, Varieties without extra automorphisms III: hypersurfaces, *Finite fields and their applications* **11** (2005), 230-268.


