Timeline for Computing 3 points Gromov-Witten invariants of the Grassmannian
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2010 at 12:54 | vote | accept | Andrea Ferretti | ||
| Apr 1, 2010 at 12:43 | answer | added | damiano | timeline score: 3 | |
| Apr 1, 2010 at 12:27 | comment | added | Andrea Ferretti | The fact that it is reduced, under these hypothesis, is a general fact about Gromov-Witten invariants, proved earlier in the book. I agree it is easy to see that there is a unique conic, I just did not think about proving it directly. I still think the hint is wrong, but the point is that, as you suggest, 3 generic lines will lie on the same ruling of the unique quadric containing them, otherwise they would meet. So in the end the curve parametrizing that ruling is the unique desired curve. If you submit this as an answer, I will be glad to accept it. By the way, are you the damiano I know? | |
| Apr 1, 2010 at 12:07 | comment | added | damiano | Reading more carefully your question, I think that in the hint, the ruled surface is meant to only contain the lines that are part of the ruling, not the spurious ones that may come when you look at the scroll in P^3. Thus in the case of degree 1, you only get the lines through the point, in the case of degree two, you only get the lines in one ruling. I thus suspect that the three lines will be in the same ruling (otherwise they would intersect, which is not very generic) and hence the unique quadric containing them will be the 1 you need. You still need to make sure it is a reduced point. | |
| Apr 1, 2010 at 11:50 | comment | added | damiano | Maybe I am missing completely the point: isn't the Grassmannian you want a quadric in P^5? And you want the conics through three given points on this quadric right? The conic will be the unique conic in the plane spanned by these three points. By homogeneity it should also be easy to show that the tangent space at this point is zero-dimensional, and thus you really should get 1 as an answer. | |
| Apr 1, 2010 at 11:07 | history | asked | Andrea Ferretti | CC BY-SA 2.5 |