Are automorphism groups of hypersurfaces reduced ? - MathOverflow

Are automorphism groups of hypersurfaces reduced ?

2010-01-04T22:08:12Z

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 ?

Answer by Bjorn Poonen for Are automorphism groups of hypersurfaces reduced ?

2010-01-05T01:07:46Z

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.