Timeline for Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| S Feb 7 at 18:05 | history | bounty ended | CommunityBot | ||
| S Feb 7 at 18:05 | history | notice removed | CommunityBot | ||
| Feb 1 at 2:07 | comment | added | naf | The statement with $\mathbb{Q}$ coefficients follows from the fact that all the data can be defined over a countable field, injectivity of the base change map with $\mathbb{Q}$ coefficients, and the fact (which you had already observed in your previous question) that the locus is closed under specialisation (which also holds with $\mathbb{Q}$ coefficients). | |
| Jan 31 at 10:32 | comment | added | Jef | Ah that's a great example. Do you have a reference for the statement with $\mathbb{Q}$-coefficients? | |
| Jan 31 at 7:34 | comment | added | naf | This is true rationally, i.e., with $\mathbb{Q}$ coefficients, but not true integrally, at least if $b$ is allowed to be any point (not necessarily closed). The problem is that increasing the base field of a variety does not induce an injection on Chow groups (but does rationally). To get an explicit example, consider the same cycle as in Jason Starr's answer to your earlier question but with $X$ there any Enriques surface over $\mathbb{C}$. Then the cycle is trivial over each closed point but not over the generic point (since there is torsion in the Neron-Severi group). | |
| S Jan 30 at 17:01 | history | bounty started | Jef | ||
| S Jan 30 at 17:01 | history | notice added | Jef | Draw attention | |
| Jan 24 at 12:13 | history | asked | Jef | CC BY-SA 4.0 |