Timeline for Computing Chow group of a variety which is almost a blow-up of another variety
Current License: CC BY-SA 4.0
5 events
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| Jul 11, 2019 at 14:33 | comment | added | A Nonny Mouse | There's a lot known about Chow rings of projective bundles (which is one way to establish the result for blow-ups, using the fact that a blowup is a subvariety of a projective bundle). You could either see if a similar result is true for your situation, or perhaps try to exploit the fact that $\pi$ is a projective bundle over $Z$. | |
| Jul 8, 2019 at 12:28 | comment | added | naf | The map $\pi_*$ in your situation is surjective. Combining this with the localisation sequence on $Y$ and the projective bundle formula (for the restriction of $\pi$ to the inverse image of $Z$) gives quite a lot of information about the Chow groups. There is probably no natural map from $CH(X)$ to $CH(Y)$ though, so it seems unlikely that there is a direct sum decomposition as in the displayed formula. | |
| Jul 8, 2019 at 8:43 | comment | added | Gro-Tsen | I studied a fairly similar situation (on an explicit case) in an arithmetic context in my paper “Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise réduction” (J. Number Theory 128 (2008), 926–944), §2 (specifically ¶2.6): I don't remember the details, which are tedious, but I remember that, while I couldn't find a general result that applied, it was possible (albeit tedious) to compute the maps explicitly on the explicit case at hand. | |
| Jul 8, 2019 at 8:25 | history | edited | Hajime_Saito | CC BY-SA 4.0 |
added 8 characters in body
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| Jul 8, 2019 at 8:16 | history | asked | Hajime_Saito | CC BY-SA 4.0 |