Timeline for What elementary problems can you solve with schemes?
Current License: CC BY-SA 2.5
16 events
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| Jan 22, 2021 at 15:37 | comment | added | Weier | @KConrad couldn't we obtain that specific result using first-order logic compactness ? (see Transfer principle, th. 3.9 in webusers.imj-prg.fr/~adrien.deloro/teaching-archive/…) | |
| Feb 4, 2019 at 8:33 | comment | added | Wojowu | @FelipeVoloch Thanks for such a quick reply! | |
| Feb 4, 2019 at 1:43 | comment | added | Felipe Voloch | @Wojowu G. Shimura, Reduction of Algebraic Varieties with Respect to a Discrete Valuation of the Basic Field, Amer. J. Math 77, (1955), 134--176. | |
| Feb 3, 2019 at 23:24 | comment | added | Wojowu | Would anyone here have pointers to Shimura's proof? I would be interested in checking it out, but couldn't find a reference. | |
| Jul 29, 2017 at 23:21 | comment | added | Noam D. Elkies | @KConrad nice example, but that curve has another "bad" prime p=3, because x^3-2 is (x-2)^3 mod 3 so there's a cusp at (x,y)=(2,0). | |
| Apr 4, 2013 at 7:23 | comment | added | Matthieu Romagny | It is especially noteworthy that "Even stating this without schemes is painful". Indeed, without schemes it is not clear what a place of bad reduction is, so that IMO this example is not quite in the family "ideally not depending on schemes" asked by the OP. | |
| Apr 4, 2013 at 1:05 | comment | added | David Corwin | +1 for the comment about Hartshorne | |
| Mar 24, 2011 at 3:22 | history | made wiki | Post Made Community Wiki by Kim Morrison | ||
| Mar 22, 2011 at 12:09 | comment | added | Hagen | Felipe Voloch's class of examples works equally well in algebraic geometry: replace $\mathrm{Spec}\mathbb{Z}$ by an (affine) algebraic curve. | |
| Mar 22, 2011 at 8:31 | comment | added | Kevin H. Lin | @KConrad, thank you for the nice explanation. | |
| Mar 22, 2011 at 4:55 | comment | added | KConrad | The word "place" is sort of a funny word for "prime". The point is that a prime for a number field can correspond to an embedding into a p-adic completion, but there are also embeddings into R or C which need to be tracked in the same way, so instead of speaking about "prime or real/complex embedding" they are all called by a neutral new word "place". | |
| Mar 22, 2011 at 4:52 | comment | added | KConrad | of depending on explicit equations. After all, maybe one equation reduces nicely mod p and another doesn't, so what should you do? With schemes you can make sense of what reduction mod p means in an equation-free way. | |
| Mar 22, 2011 at 4:51 | comment | added | KConrad | Kevin, it's a prime p where the reduced curve mod p is not smooth. So the reduction is "bad" in the sense that it is geometrically worse than what you started with. For example, consider the cubic curve y^2 = x^3 - 2, which is smooth in characteristic 0. When you reduce it mod p it is still smooth except when p = 2, in which case it turns into the singular curve y^2 = x^3 (cusp at origin). So we'd say 2 is a prime (place) of bad reduction. Actually, reducing curves mod p is a setting where schemes are a useful language, because it lets you discuss reduction without the crutch (continued...) | |
| Mar 22, 2011 at 4:31 | comment | added | Kevin H. Lin | For us non-arithmetic geometers, can you explain what a "place of bad reduction" is? | |
| Mar 22, 2011 at 3:22 | comment | added | Yemon Choi | Knowing as little about algebraic geometry as I do, I like the sound of this answer since it seems close in spirit to the example mentioned by the original poster (Burnside's proof of $p^aq^b$ via character theory). | |
| Mar 22, 2011 at 3:18 | history | answered | Felipe Voloch | CC BY-SA 2.5 |