Timeline for The pseudoeffective cone does not contain lines
Current License: CC BY-SA 3.0
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| Oct 27, 2015 at 18:38 | comment | added | David E Speyer | @JasonStarr Just to check, your comment is about effective divisor classes, right? Because everything else in the question and Michael's answer applies to effective classes in any dimension, but I believe that Nakajima's smooth proper nonprojective 3-fold has lines in the cone of effective curve classes. | |
| Jun 13, 2012 at 15:04 | comment | added | Jason Starr | Properness is sufficient. By Chow's Lemma, there exists a projective, birational morphism from a projective variety to the original variety. Under pullback by a birational morphism, effectivity is preserved (hence so is pseudo-effectivity). So for a class $\beta$ as above, the pullback is numerically trivial. In other words, for every curve in the blowup, the image curve in the original variety has $0$ pairing with $\beta$. However, every integral curve in the original variety is the (finite-to-one) image of a curve in the blowup. So $\beta$ is numerically trivial. | |
| Jun 13, 2012 at 9:49 | vote | accept | fds | ||
| Jun 12, 2012 at 15:44 | history | edited | Michael Joyce | CC BY-SA 3.0 |
added 118 characters in body
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| Jun 12, 2012 at 15:42 | history | undeleted | Michael Joyce | ||
| Jun 12, 2012 at 15:38 | history | deleted | Michael Joyce | ||
| Jun 12, 2012 at 15:32 | history | answered | Michael Joyce | CC BY-SA 3.0 |