Timeline for Are automorphism groups of hypersurfaces reduced ?
Current License: CC BY-SA 2.5
16 events
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| Jan 6, 2010 at 0:26 | comment | added | algori | Olivier -- yes, I was thinking about the projective automorphism groups. Thanks for mentioning this, I definitely have to take a look in that book. | |
| Jan 5, 2010 at 12:21 | comment | added | Olivier Benoist | @algori : In fact, we were both right : the automorphism group of elliptic curves in char $3$ is reduced, but their projective automorphism group isn't. It is well explained in the book of Katz and Sarnak to which Bjorn Poonen refers in his answer. | |
| Jan 5, 2010 at 2:06 | comment | added | VA. | @Pete: I should have said if $d\ge3$ + $X$ is NOT an elliptic curve then there are no infinitesimal automorphisms (projective or not). And if $\dim X\ge3$ then $Pic X=Z$, so all automorphisms are projective (since $O(1)$ goes to itself). | |
| Jan 5, 2010 at 1:28 | vote | accept | Olivier Benoist | ||
| Jan 5, 2010 at 1:22 | comment | added | algori | Pete -- Calabi-Yau hypersurfaces of dimension 3 or more have finite automorphism groups, see my comment above. Moreover, all their automorphisms are projective. | |
| Jan 5, 2010 at 1:07 | answer | added | Bjorn Poonen | timeline score: 17 | |
| Jan 5, 2010 at 1:02 | comment | added | Pete L. Clark | @VA: Okay, I checked -- you are talking about projective automorphisms. If I recall correctly, the point is that the condition $d > N+1$ forces all automorphisms to be projective, and I think this is true regardless of characteristic. To clarify my original comment: I suppose things like Calabi-Yaus of arbitrary dimension fall under "known exceptions". | |
| Jan 5, 2010 at 0:57 | comment | added | Pete L. Clark | @VA: I must not be understanding you correctly, since elliptic curves have infinite automorphism group, and I believe that there are Calabi-Yau hypersurfaces of every dimension with infinite automorphism group. Do you perhaps mean projective automorphisms? (I didn't.) | |
| Jan 5, 2010 at 0:16 | comment | added | VA. | @Pete: No, the sufficient condition is $d\ge3$, see e.g. Mumford GIT 4.2, where the reference is to a Kodaira-Spencer's paper (which I am guessing is not helpful for char p). | |
| Jan 5, 2010 at 0:09 | comment | added | Olivier Benoist | @Algori : I don't think it will work : for an elliptic curve, $T_X$ is trivial, so the automorphism group has tangent space of dimension $1$ at the identity. Hence the (connected component of the) automorphism group has to be the reduced elliptic curve itself. | |
| Jan 5, 2010 at 0:07 | comment | added | Olivier Benoist | @Pete : It is true that varieties of general type have finite automorphism groups, but the converse is not (think of a blow-up of the plane in more than four points !). And for hypersurfaces, this happens most of the time (example : cubic surfaces which are blow-ups of the plane in six points). However, as long as I know, varietes of general type could very well have nonreduced automorphism groups (?). I don't have any example at hand. | |
| Jan 4, 2010 at 23:58 | comment | added | algori | Olivier -- I'm not 100% sure, but you could try cubic curves over a field of characteristic 3 to get a non-reduced example. But in any characteristic the projective automorphism groups are finite. | |
| Jan 4, 2010 at 23:55 | history | edited | Olivier Benoist | CC BY-SA 2.5 |
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| Jan 4, 2010 at 23:53 | comment | added | algori | The projective automorphism group of a hypersurface is finite, except for quadrics. The whole automorphism group coincides with the projective one except in the cases mentioned by Olivier. But this happens for different reasons if $N>2,d\neq N+1$ and if $N>3$. Namely, if $N>2$, then we can use the fact that the Picard is torsion free and the canonical class is $n+1$ times the hyperplane section. If $N>3$, then the Picard is $\mathbf{Z}$ and the effective cone is spanned by the hyperplane section. | |
| Jan 4, 2010 at 23:13 | comment | added | Pete L. Clark | Could you clarify what you mean by "known exceptions"? So far as I know, a necessary and sufficient condition for the automorphism group of every smooth degree $d$ hypersurface in $\mathbb{P}^N$ [over a fixed, say, algebraically closed field] to be finite is $d > N+1$, equivalently for the variety to be of general type. I also thought that it was a general theorem that varieties of general type had finite automorphism group: is this harder to prove in characteristic $p$? (Is it false?!?) | |
| Jan 4, 2010 at 22:08 | history | asked | Olivier Benoist | CC BY-SA 2.5 |