Timeline for A weak version of high dimensional Abhyankar's conjecture
Current License: CC BY-SA 3.0
12 events
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| Aug 3, 2017 at 15:49 | comment | added | Ariyan Javanpeykar | @PiotrAchinger Let Et be the stack of finite etale morphisms. (So objects are $(S,f:X\to S)$ with $S$ a scheme and $f$ finite etale.) This is an algebraic stack locally of finite type. Let Et$_d$ be the substack of finite etale morphisms of degree $d$. This is of finite type. You are right that $\mathrm{Hom}(\mathbb A^1, \mathrm{Et})$ is not representable by an algebraic stack. (Your argument shows that Artin's axioms fail. Morally, this is because $\mathbb A^1$ is not proper.) | |
| Jun 26, 2017 at 16:15 | vote | accept | John Z. | ||
| Jun 25, 2017 at 22:10 | history | edited | YCor | CC BY-SA 3.0 |
fixed typo in title
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| Jun 25, 2017 at 21:02 | comment | added | Piotr Achinger | @AriyanJavanpeykar The situation in characteristic $p$ is quite perplexing: any $f\in A$ gives rise to the Artin-Schreier $\mathbb{Z}/p$-covering $t^p - t = f$ of $X={\rm Spec} A$, and $f$ and $g$ yield the same covering iff $f-g = h^p - h$ for some $h\in A$. In particular, unlike in char. 0, one can easily have "families" of coverings, e.g. $t^p-t=xy$, as a family of coverings of $\mathbb{A}^1$ parametrized by $\mathbb{A}^1$. On the other hand, by the unique lifting property of etale maps, every such family is formally constant. This seems to mean that there is no reasonable moduli problem. | |
| Jun 25, 2017 at 20:55 | answer | added | Piotr Achinger | timeline score: 7 | |
| Jun 25, 2017 at 20:49 | comment | added | Ariyan Javanpeykar | @PiotrAchinger You're right. What I wrote only works when the scheme is generically proper over $A$. My bad. | |
| Jun 25, 2017 at 20:44 | comment | added | Piotr Achinger | @AriyanJavanpeykar Why is the stack of finite type? Here $X$ is affine, and if we are in characteristic $p$, we probably do not even have such a stack, and even if we did, the moduli problem would be unbounded... | |
| Jun 25, 2017 at 20:43 | comment | added | John Z. | Yes, I do assume that Galois coverings are finite etale coverings, and variety means integral scheme of finite type. | |
| Jun 25, 2017 at 20:18 | comment | added | Ariyan Javanpeykar | ....In particular, any $k$-point of characteristic $p>0$ (with $k$ the residue field of a maximal ideal of $A$) lifts to characteristic zero (and thus to a $\mathbb C$-point). So, if you have a finite etale cover of $X_k$, it will "come" from a finite etale cover of $X$ (via specialization). | |
| Jun 25, 2017 at 20:17 | comment | added | Ariyan Javanpeykar | When you say "variety" over $S$, do you mean "scheme"? Also, is your Galois covering (finite) etale? If yes, for $d\geq 1$, let $M_d$ be the stack of finite etale covers of $X$ over $A$. This is a finite type stack over $A$. Like "everything" over $A$, it is generically flat over the function field $K(A)$ of $A$. Therefore, by spreading out (see Rydh's appendix in arxiv.org/abs/0904.0227), replacing Spec $A$ by a dense open if necessary, we may and do assume that $M_d$ is flat over $A$.... | |
| Jun 25, 2017 at 19:29 | review | First posts | |||
| Jun 25, 2017 at 19:31 | |||||
| Jun 25, 2017 at 19:26 | history | asked | John Z. | CC BY-SA 3.0 |