Timeline for Codimension restrictions on intersections
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2018 at 21:40 | comment | added | Jason Starr | If you have a family of cycles that is proper over the base, then the intersection of that family of cycles with the closed subset $Z$ is also proper over the base. Then you can apply Chevalley's theorems about upper semicontinuity of fiber dimension to prove openness of the locus in the base over which the fibers have dimension less than a specified integer. | |
| Jan 14, 2018 at 21:32 | comment | added | Jason Starr | Koll'ar's Chow functor is defined to be the functor of effective cycles that are also proper over the base. | |
| Jan 14, 2018 at 21:26 | comment | added | user113452 | @JasonStarr Upon assuming the characteristic of $k$ is zero, Kollar's Chow functor is representable in the quasi projective case too (Thm. 5.4 loc cit). I agree with your argument about the non openness aspect. | |
| Jan 14, 2018 at 15:22 | comment | added | Jason Starr | If $X$ is not proper, there typically is no "Chow variety of effective cycles" in $X$. The Chow functor (say, on the category of seminormal, separated, finite type $K$-schemes as in the book Rational curves on algebraic varieties by Koll'ar) is not representable. You could just ask your question for some particular family of cycles over a specified base scheme. However, the answer is negative for the same reason as in the other comment that I wrote today. | |
| S Jan 14, 2018 at 13:43 | history | suggested | user119470 |
new tag added
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| Jan 14, 2018 at 13:42 | review | Suggested edits | |||
| S Jan 14, 2018 at 13:43 | |||||
| Jan 14, 2018 at 13:36 | history | asked | user113452 | CC BY-SA 3.0 |