Timeline for Number of lines incident to a fixed line on quartic threefolds
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 9, 2020 at 15:39 | comment | added | Jason Starr | @Sasha You are correct -- I should have subtracted the degree on the line of the normal bundle of the (normalization) of the scroll. | |
| Mar 8, 2020 at 20:14 | comment | added | AG learner | @Sasha Thanks again for suggesting the reference. I traced back to Tyurin's Five lectures on three-dimensional varieties and found a proof there. | |
| Mar 8, 2020 at 20:12 | answer | added | AG learner | timeline score: 2 | |
| Mar 7, 2020 at 8:21 | comment | added | Sasha | @JasonStarr: In fact, when you count $n_L$ you should also count $L$ with multiplicity $-1$, so the actual number of lines intersecting $L$ and distinct from $L$ is 81. | |
| Mar 7, 2020 at 8:19 | comment | added | Sasha | @AGlearner: You can also look into "Markushevich, D. G. Numerical invariants of families of lines on some Fano varieties. Mathematics of the USSR-Sbornik, 1983, 44:2, 239–260", where similar computations with more explanations for other Fano threefolds are performed. In partiuclar, Lemma 2.6 explains how to go from the number 320 to the number $n_L$. | |
| Mar 6, 2020 at 22:23 | comment | added | AG learner | @JasonStarr Thanks for the hint! I learned from Tennison's paper how to get the number 320. This can be also interpreted as the number of lines in $X$ incident to a general plane, but how did you get $n_L=80$ from this? Thanks again for your help! | |
| Mar 6, 2020 at 22:16 | comment | added | AG learner | @Sasha Thanks for the nice reference! Now I am thinking the following way to compute: Let $G=Gr(2,5)$, and $I\subset G\times G$ be the correspondence that parameterizes pair of incident lines. Let $p,q$ be coordinate projection. I think $n_L$ should be $p^*F\cdot I\cdot q^*F$. Is that right? Project it to the first coordinate, it should be $F\cdot p_*(I\cdot q^*F)$. I know $F=32\sigma_1\sigma_{1,1}(3\sigma_1^2+4\sigma_{1,1})$ from Tennison's paper, but I don't know how to do the rest into Schubert cycles. Could you help me on this? | |
| Mar 5, 2020 at 19:34 | comment | added | Sasha | The standard reference is "Tennison, B. R. On the quartic threefold. Proc. London Math. Soc. (3) 29 (1974), 714–734." | |
| Mar 5, 2020 at 17:37 | comment | added | Jason Starr | The degree of the surface equals $320$, so I believe $n_L$ equals $80$. | |
| Mar 5, 2020 at 16:20 | history | asked | AG learner | CC BY-SA 4.0 |