Timeline for problematic proof of the moving lemma, second part ?
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| Aug 20, 2017 at 17:20 | comment | added | R. van Dobben de Bruyn | There is another problem with using the moving lemma to define intersection products, and that is well-definedness. That is to say, if $\alpha, \beta$ are given, and $\alpha'$ and $\alpha''$ are rationally equivalent to $\alpha$ and meet $\beta$ properly, then one should check that $\alpha' \cdot \beta = \alpha''\cdot \beta$. Moreover, one wants the intersection product to factor through rational equivalence, so one should vary $\beta$ as well. This method is used in the Stacks Project, and all details are carefully carried out there. | |
| Dec 17, 2012 at 23:25 | comment | added | nono | By the way, the first part says that you can find a cycle W" which is rationally equivalent to W' such that W.W" is well-defined. If you accept the first part, then I think you must accept the second part as well, at least in principle. This is because to say that W" is rationally equivalent to W', you must have a family W(t) with $t\in \mathbb{P}^1(k)$ so that $W(0)=W'$ and $W(\infty )=W"$. Now this family intersects properly with $W$ at one point $\infty$, it should do so for a dense open set in $\mathbb{P}^1$. | |
| Dec 17, 2012 at 23:10 | comment | added | nono | If the story goes as you said, then there must be some problem. But since you don't know what problem is, it is hard to say. Anyway, I still think that the case when X is smooth and k is algebraic closed, there is no problem. There is a book of Eisenbud and Harris, which dates 27 April 2010 isites.harvard.edu/fs/docs/icb.topic720403.files/book.pdf which discusses Chow moving lemma in Chapter 2 for X smooth quasi-projective and k algebraic closed. I guess the date should be later than the story you referred to, so there would not be any problem in this case. | |
| Dec 17, 2012 at 18:19 | comment | added | Andrew | Nono, thanks. I do not really know, I've only heard these claims but was unable to clarify what is meant...I think that at some point someone convinced Suslin that there is a problem with published proofs, but I do not know in what generality etc, and that was few years back (but after the paper you refer to). | |
| Dec 17, 2012 at 17:42 | comment | added | nono | Thanks, Andrew. However, I still do not know what parts of the proof that you think are problematic. Did the experts you contacted say so? Do you consider X not be smooth, or do you consider the case where the field k is not algebraic closed, or else? There is also the paper "Moving algebraic cycles of bounded degree" on Inventiones math. by Friedlander and Lawson, in which they present a proof of the result in Roberts as well (see page 99 on in the paper). | |
| Dec 17, 2012 at 16:30 | comment | added | Andrew | Nono, thanks, I corrected the statement. I should have been more clear: the claim I heard was that there is no complete proof in the literature, or rather that some experts think so. | |
| Dec 17, 2012 at 16:29 | history | edited | Andrew | CC BY-SA 3.0 |
a misprint wrt nono's comment
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| Dec 17, 2012 at 15:23 | comment | added | nono | I think that the proof of Roberts is correct for the case where the ground field is algebraic closed. May be the gap, if any, happens when k is not algebraic closed. By the way, in the second part, the family Z(t) is rational ($t\in \mathbb{P}^1(k)$), which is stronger than that you stated. | |
| Dec 17, 2012 at 13:30 | comment | added | naf | This is not the reason that the moving lemma is not used in Fulton's book. The way intersection theory is developed there is more general, since it requires no quasi-projectivity assumptions, and much more powerful, since he constructs a "refined" intersection product which is impossible using the moving lemma. | |
| Dec 17, 2012 at 13:08 | history | asked | Andrew | CC BY-SA 3.0 |