Timeline for Are projective toric varieties, locally complete intersection?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2011 at 17:12 | vote | accept | Mohammad Farajzadeh-Tehrani | ||
| Apr 12, 2011 at 20:18 | comment | added | Karl Schwede | @Mohammed, (Gorenstein Toric => LCI?) I suspect the answer is no. See exercise 21.3 in Matsumura's book. That ring is not normal, but it's really close to being toric (it is a binomial ring). You could try googling things like "affine semi-group ring" (commutative algebraists version of toric) and "complete intersection", I suspect there are examples in the literature which explain when this happens. | |
| Apr 12, 2011 at 20:11 | comment | added | roy smith | The point seems to be that being an lci under one closed embedding in a smooth ambient guarantees it for all such embeddings. And that is one definition of an lci variety. | |
| Apr 12, 2011 at 20:10 | comment | added | Karl Schwede | @J.C. Ottem, I don't think it does depend on the embedding, see for example Chapter 21 of Matsumura's commutative ring theory. There they basically define a local ring to be an LCI if its completion is an LCI (in the usual sense). However, this is done because not all rings are quotients of regular rings. But by Theorem 21.1(iii), for an LCI which is a quotient of ANY regular ring, it can be cut out by the right number of generators. | |
| Apr 12, 2011 at 20:02 | comment | added | Mohammad Farajzadeh-Tehrani | What about my main question: Do you think we can prove that projective Gorenstein Toric varieties are l.c.i. You can even assume it is Fano, if it helps. | |
| Apr 12, 2011 at 19:43 | comment | added | J.C. Ottem | Yes, I was talking about a general ambient variety. If you mean an embedding in projective space, this is probably true. | |
| Apr 12, 2011 at 19:37 | comment | added | mdeland | yes, being (an) lci (morphism) is independent of the choice of factorization into something smooth. This is definitely discussed in Fulton's book on intersection theory, for example (and no doubt can be found in sga | |
| Apr 12, 2011 at 18:56 | comment | added | roy smith | I am not expert but this seems to contradict the Rmk. on page 185 of Hartshorne that being an lci in a smooth variety is a property independent of the embedding, and that one proves it using the cotangent complex of Lichtenbaum and Schlessinger. | |
| Apr 12, 2011 at 18:16 | vote | accept | Mohammad Farajzadeh-Tehrani | ||
| Apr 13, 2011 at 17:12 | |||||
| Apr 12, 2011 at 18:16 | comment | added | Mohammad Farajzadeh-Tehrani | Of course. I meant: Is a projective toric Gorenstein variety local complete intersection? do you mean any Gorenstein projective variety is l.c.i? Also I had this question: does being a l.c.i depends on particular choice of embedding? | |
| Apr 12, 2011 at 18:12 | comment | added | J.C. Ottem | A variety is Gorenstein if it is Cohen-Macaulay and it's canonical divisor is a line bundle. These properties does not depend on the embedding. | |
| Apr 12, 2011 at 18:08 | comment | added | Mohammad Farajzadeh-Tehrani | In fact I am interested in Gorenstein, toric varieties. What can we say then? | |
| Apr 12, 2011 at 18:01 | history | answered | J.C. Ottem | CC BY-SA 3.0 |