Timeline for Gromov-Witten invariants counting curves passing through two points
Current License: CC BY-SA 2.5
5 events
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| Feb 19, 2018 at 15:52 | comment | added | user74900 | Dear Zhiyu, is $\langle [pt], \dots \rangle^{X}_{0, C}$ a statement? I, being a non-expert, think that it is a number so that '...equivalent to $\langle [pt], \dots \rangle^{X}_{0, C}$' does not make much sense. | |
| Dec 21, 2010 at 16:28 | comment | added | Zhiyu | Sorry but you need to be a little bit more careful about the bend-and-break. It only works if you can deform the elliptic curve with a fixed complex structure. But if you can vary the complex structure, then it will specialize to a nodal rational curve. In this case you are still fine. | |
| Dec 21, 2010 at 15:55 | comment | added | Zhiyu | In case of $g=1$, the condition implies that there is a curve of arithmetic genus 1 passing through 2 general points. If this is a irreducible embedded curve, then you can deform it with 1 point fixed. Then bend and break tells you that there is a rational curve through the fixed point. If this is not an irreducible smooth embedded curve, then there are components of genus 0, passing through 1 of the two general points. In any case, there is a rational curve through a general point. In general, I think you need a genus g curve with g+1 points. | |
| Dec 21, 2010 at 2:44 | comment | added | Mike Usher | Thanks, Zhiyu! I wasn't aware of the fact that the condition when g=1 implied uniruledness. Is there a simple explanation for why this is? Any reason to think it might be true for higher genera? | |
| Dec 20, 2010 at 19:32 | history | answered | Zhiyu | CC BY-SA 2.5 |