Timeline for Is there a "geometric" intuition underlying the notion of normal varieties?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Feb 16, 2015 at 4:37 | history | edited | Karl Schwede | CC BY-SA 3.0 |
added reference to related answer
|
| Oct 23, 2013 at 20:04 | comment | added | Karl Schwede | I can also explain the details to you in person when you visit PSU next month. | |
| Oct 23, 2013 at 11:47 | comment | added | Karl Schwede | @JoshuaGrochow I don't think I do know a published reference. I could try to write details here if you really need a reference. | |
| Oct 23, 2013 at 1:38 | comment | added | Joshua Grochow | @KarlSchwede: Do you know a reference to the fact that any non-normal variety can be gotten by this gluing construction? | |
| Oct 25, 2012 at 4:43 | comment | added | Karl Schwede | Very possibly, I'd have to think about it. | |
| Oct 25, 2012 at 2:22 | comment | added | roy smith | so maybe we can see the induced map of tangent cones in that example, from the cone over the curve in P^4 to the cone over the curve in P^3? | |
| Oct 24, 2012 at 11:33 | comment | added | Karl Schwede | So, the other way to break non-S2-ness, besides gluing points, is to kill tangent information at a point. That is what's going on with the first example (cone over the quartic rational curve in $\mathbb{P}^3$) although I must admit, I don't have a good way to visualize that example But generally, the easiest way to make a non-normal graded ring is to kill some low degree terms in a normal graded ring. This will make something that is not S2 (unless the graded ring has dimension $\leq 1$). | |
| Oct 24, 2012 at 6:08 | comment | added | roy smith | ok, i'm going to say that i am back to mumford's point, i.e. the only finite birational maps that are (to me) obviously not isomorphisms are those that are not bijective. so the only case where i can really see geometrically that a variety is not normal is one that is not unibranch. then its normalization is not bijective. so this is a little larger class than "not R1", but not fully the class of "not S2". i.e. I can really see geometrically that a non unibranch variety is not normal, but it is not so easy to see geometrically that a variety is not S2. | |
| Oct 24, 2012 at 2:50 | comment | added | roy smith | Here is an example I find hard to visualize fully geometrically by the method I callee "geometric": take a smooth rational quartic curve in P^3 and let X be the cone over it in P^4. This seems to be a standard example of a non normal surface satisfying R1 but not S2. One can use geometry to give a finite birational map to X (I hope), (by projecting the cone in P^5 over a rational quartic curve in P^4 onto X), but is there a "geometric" way to see this map is non trivial? My feeling now is that S2, i.e. depth, is hard to make fully geometric. | |
| Oct 13, 2012 at 2:36 | history | edited | Karl Schwede | CC BY-SA 3.0 |
edited body
|
| Oct 13, 2012 at 0:03 | comment | added | roy smith | very nice job. much more pregnant and also more clear and precise than my comment. this puts flesh on it and cries out for exploration. | |
| Oct 12, 2012 at 19:57 | history | answered | Karl Schwede | CC BY-SA 3.0 |