Timeline for Non-simply-connected smooth proper scheme over Z?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2012 at 8:38 | comment | added | Christian Liedtke | ... of course, these effects were known to Igusa and Serre already, but these examples are maybe more explicit | |
| May 16, 2012 at 8:37 | comment | added | Christian Liedtke | For example, William Lang ('Classical Godeaux Surface in Characteristic p', Math. Ann. 256, 419-427 (1981)) and Rick Miranda ('Nonclassical Godeaux Surfaces in Characteristic Five', Proc. AMS 91, 9-11 (1984)) address this. I'm not sure whether their examples are defined over the integers as they choose generic hyperplane sections, but maybe a computer search can establish this by hand? Miranda's examples also show explicitly that the Picard scheme need not be smooth - it becomes non-reduced for p=5 (in fact ${\rm Pic}^0$ is isomorphic to the non-reduced scheme $\mu_5$) ... | |
| May 16, 2012 at 4:13 | history | edited | Ben Wieland | CC BY-SA 3.0 |
retraction
|
| May 16, 2012 at 2:31 | comment | added | Will Sawin | Well then you should probably make that your answer to this question. | |
| May 16, 2012 at 2:02 | comment | added | Ben Wieland | The Atiyah-Bott fixed point formula plus the obvious (conjectural) generalization of Fontaine's result shows that a smooth proper variety over Z must have trivial (pro-finite) fundamental group. Fontaine's proven theorem covers the case of dimension 1-3. | |
| May 15, 2012 at 17:36 | comment | added | Ben Wieland | I just meant that since there is a standard conjecture that smooth and proper over Z implies rational, that nonnegative Kodaira dimension would be a more surprising counter-example. Fontaine's letter to Messing probably already forces rationality in dimension 2. ams.org/mathscinet-getitem?mr=1274493 | |
| May 15, 2012 at 3:31 | comment | added | Will Sawin | If there's some good reason why negative Kodaira dimension is necessary for smoothness, Castlenuovo's theorem on the generic fiber would imply that all dimension 2 proper smooth surfaces over Spec Z are rational, thus simply connected, pushing the minimum possible dimension of a counterexample up to 3. | |
| May 15, 2012 at 2:41 | comment | added | Will Sawin | I was trying to construct a similar argument. I guessed the step "nonnegative Kodaira dimension => nonsmooth" purely on blind hope. Why is that true? (or implausibly untrue?) For n>1 all n monomials must have minimum degree p, else there are not enough monomials to avoid all the nonsingular points. You get the same computation for the Kodaira dimension nonnegative. | |
| May 15, 2012 at 0:58 | comment | added | Ben Wieland | Oh, right, unbounded degree is bad for smoothness over $Z$. Since it doesn't work to produce a free action on a curve, it probably doesn't work to produce a free action on a simply connected space. More specifically, in the case $n=1$, avoiding the fixed point at $p$ requires an invariant monomial, thus minimum degree $p$, yielding nonnegative Kodaira dimension, making it implausible that it be smooth. | |
| May 13, 2012 at 7:42 | comment | added | Will Sawin | Getting polynomials smooth over every $p$ might be tricky. It doesn't seem like the sort of thing that is impossible, but, for instance, you can't do it in dimension $2$ even with arbitrarily high degree. Do you think it would be possible to explicitly work out an example for some small $p$ and $n$, to show it exists? | |
| May 13, 2012 at 5:59 | comment | added | Ben Wieland | Yes, $G$-invariant polynomials; or, equivalently, work downstairs. The degree is a free parameter. The higher degree, the larger the space to choose from, avoiding any problematic space of fixed dimension. | |
| May 13, 2012 at 5:42 | comment | added | Will Sawin | What do you mean by "generic complete intersection"? What degree polynomials are you intersecting? Are they intended to be $G$-invariant polynomials? | |
| May 13, 2012 at 4:39 | history | answered | Ben Wieland | CC BY-SA 3.0 |