Are automorphism groups of hypersurfaces reduced ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:33:40Z http://mathoverflow.net/feeds/question/10743 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10743/are-automorphism-groups-of-hypersurfaces-reduced Are automorphism groups of hypersurfaces reduced ? Olivier Benoist 2010-01-04T22:08:12Z 2010-01-05T01:14:38Z <p>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 ?</p> <p>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 ?</p> http://mathoverflow.net/questions/10743/are-automorphism-groups-of-hypersurfaces-reduced/10759#10759 Answer by Bjorn Poonen for Are automorphism groups of hypersurfaces reduced ? Bjorn Poonen 2010-01-05T01:07:46Z 2010-01-05T01:14:38Z <p>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, <em>Random matrices, Frobenius eigenvalues, and monodromy</em>, 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 <a href="http://www-math.mit.edu/~poonen/papers/projaut.pdf" rel="nofollow">http://www-math.mit.edu/~poonen/papers/projaut.pdf</a> , which is Poonen, Varieties without extra automorphisms III: hypersurfaces, <em>Finite fields and their applications</em> <strong>11</strong> (2005), 230-268.</p>