Timeline for Finiteness property of automorphism scheme
Current License: CC BY-SA 2.5
23 events
| when toggle format | what | by | license | comment | |
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| Sep 23, 2016 at 1:59 | answer | added | John L | timeline score: 17 | |
| Apr 30, 2010 at 17:35 | vote | accept | BCnrd | ||
| Apr 30, 2010 at 17:35 | history | bounty ended | BCnrd | ||
| Apr 26, 2010 at 14:01 | answer | added | Torsten Ekedahl | timeline score: 16 | |
| Apr 25, 2010 at 17:48 | comment | added | Torsten Ekedahl | @VA: I am not sure what you mean. Are you saying that any possible ample cone can be obtained by choosing a torsion point? Also to avoid confusion by general point I mean one which lies outside all proper closed subgroups. | |
| Apr 25, 2010 at 15:32 | comment | added | VA. | Torsten: good point. Blowing up general/nongeneral points in Zariski topology changes the ample cone, but torsion points are dense so they are general. So I don't see how to make this idea work. | |
| Apr 25, 2010 at 15:22 | comment | added | Torsten Ekedahl | @VA again: How does that work for non-minimal surfaces? Does the ample cone (or whatever) suffice to see the difference between blowing up two general points on an abelian surface (one must blow up two of them not one as I said) which gives trivial automorphism group and blowing up two whose difference is torsion in the group structure which gives a group commensurable to the group of the abelian surface itself? | |
| Apr 25, 2010 at 14:46 | comment | added | VA. | I polled some colleagues and J. Koll'ar pointed out that the version with (at least simply connected) Kahler variety and Diff(X) instead of Aut(X) is true, by D. Sullivan "Infinitesimal computations in topology", Thm.12.4. (The group of components of Diff(X) is commensurable to the Aut of $H^*(X,Z)$ preserving Pontryagin classes). He suggested that it might be helpful to compare Aut(X) with $Aut(H^*(X,Z))$ preserving the geometric package: Chern classes, Hodge structures (and also effective classes, ample classes...). | |
| Apr 25, 2010 at 14:42 | comment | added | VA. | @@Torsten: you are right about the K3s. I will edit my comment and attach it below; sorry for the little discontinuity this creates. | |
| Apr 25, 2010 at 6:22 | comment | added | Torsten Ekedahl | (cont'd) Also taking a general point on the product of an elliptic curve with itself and blowing it up will have trivial automorphism while the automorphism group of the Hodge structure (fixing Chern classes) will be equal to the automorphism group of the product which is infinite. | |
| Apr 25, 2010 at 6:17 | comment | added | Torsten Ekedahl | @VA: Isn't the situation more complicated already for K3-surfaces? In that case the arithmetic group you are talking about is commensurable with the semi-direct product of the normal subgroup generated by reflections in $-2$-curves and the actual automorphism group of the K3-surface and there are examples when the normal subgroup has finite index. It is true that which elements of $H^\ast(X,\mathbb Z)$ are $-2$-curves is determined by the Hodge structure so that the automorphism group is determined by Hodge structure data. | |
| Apr 24, 2010 at 16:52 | comment | added | Torsten Ekedahl | Dolgachev: Infinite Coxeter groups and automorphisms of algebraic surfaces I just found by searching in SciMath and haven't looked at it. I think the crucial case is probably K3-surfaces and abelian varieties which can be handled by periods. An interesting further case would be to look at K3-surfaces and consider a stabiliser of a point. That will be the automorphism group of the blow up in that point and could possibly be non-f.g. I don't think that can be handled by period theory however. | |
| Apr 24, 2010 at 16:51 | comment | added | naf | A reference is the article "Reflection groups in algebraic geometry" by Dolgachev in the Bulletin of the AMS. | |
| Apr 24, 2010 at 16:39 | history | bounty started | BCnrd | ||
| Apr 23, 2010 at 16:28 | comment | added | BCnrd | @Torsten: can you provide a reference (e.g., title?) for the paper of Dolgachev which you mention above? Thanks. | |
| Apr 23, 2010 at 5:15 | comment | added | Chandan Singh Dalawat | "Mazur wrote a paper ... " On the passage from local to global in number theory. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 14--50. | |
| Apr 23, 2010 at 3:53 | comment | added | BCnrd | VA, Mumford gave a colloq. talk here today, and after the dinner I mentioned the question. He was intrigued, and said he'd never heard anything about a result in that direction. Mazur wrote a paper with some theoretical arguments for Tate-Shaf. sets which required finite presentation hypotheses on Aut-scheme. In that part he acknowledged assistance from Gabber, to the extent of saying that for some result Gabber weakened the hypothesis to finite generation...but not eliminated it! So seems Gabber thought about it without success. If it is known, I will then be amazed (and very happy). | |
| Apr 22, 2010 at 21:32 | comment | added | VA. | Right. One can look at other features of NS(X) that are preserved. For example for Fanos, the closure of the ample cone is finitely generated, so that leads to the proof. CYs seem like maybe the hardest case. Such a natural question... Must be known, I hope someone answers. | |
| Apr 22, 2010 at 17:53 | comment | added | BCnrd | VA, the question of the image of Aut(X) in Aut(NS(X)) was the only idea I ever had on this, and I don't know any device to control the image. Since GL_n(Z) has subgroups which aren't finitely generated, it then seems to hit a brick wall (unless there's another idea one can bring in about NS(X)). | |
| Apr 22, 2010 at 17:53 | comment | added | Torsten Ekedahl | For minimal surfaces a result of Dolgachev says that (possibly over the complex numbers only) that the image of $\mathrm{Aut}(X)$ in $\mathrm{Aut}(K_X^\perp)$ (the orthogonal complement of the canonical class) is a quotient of a subgroup of finite index of the full automorphism group of that lattice. Hence it is at least finitely generated. The normal subgroup by which one takes the quotient is the subgroup generated by reflections in nodal curves. | |
| Apr 22, 2010 at 17:40 | comment | added | VA. | Fix an ample class $L$ in Neron-Severi group of $X$. The subgroup of automorphisms sending $L$ to itself is of finite type. So the real question is: what is the image of $Aut(X)$ in $Aut(NS(X))$? Is that finitely generated/presented? These automorphisms permute ample classes, so if the semigroup of ample classes is f.g. (which happens rarely), we are OK. | |
| Apr 22, 2010 at 16:23 | history | edited | BCnrd | CC BY-SA 2.5 |
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| Apr 22, 2010 at 16:07 | history | asked | BCnrd | CC BY-SA 2.5 |