Timeline for Picard groups and birational morphisms
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 23, 2016 at 18:20 | vote | accept | CommunityBot | ||
| Nov 22, 2016 at 17:10 | answer | added | Sándor Kovács | timeline score: 2 | |
| Nov 22, 2016 at 10:57 | comment | added | Jason Starr | @Jessica_90. "If $Y$ is singular just in points and the singularities come from contractions via $f$ of smooth subvarieties of $X$ of codimension greater or equal than two do we know that $Y$ is locally factorial?" If there is a contracted subvariety $Z$ of codimension greater than or equal to two, then you know that $Y$ is not locally factorial. I believe this result is originally due to Abhyankar (but I cannot find the reference). I learned this in Debarre's wonderful textbook. In the following link, it is Prop. 8.12, p. 85, math.ens.fr/~debarre/M2.pdf | |
| Nov 21, 2016 at 20:56 | comment | added | user97096 | Thanks. If $Y$ is singular just in points and the singularities comes from contractions via $f$ of smooth subvarieties of $X$ of codimension greater or equal than two do we know that $Y$ is locally factorial ? | |
| Nov 21, 2016 at 17:52 | comment | added | Jason Starr | You should assume that $Y$ is locally factorial if you want this to hold. There are criteria for local factoriality in SGA 2, and there are other criteria that come out of the Minimal Model Program. | |
| Nov 21, 2016 at 15:36 | comment | added | user97096 | Sure we may assume that $X$ is smooth and $Y$ is normal. But I guess this is not enough, take for instance Y a quadric cone of dimension two. | |
| Nov 21, 2016 at 15:33 | comment | added | abx | What are your hypotheses on the singularities of $X$ and $Y$? You should at least assume that $Y$ is normal. | |
| Nov 21, 2016 at 15:29 | history | asked | user97096 | CC BY-SA 3.0 |