Timeline for Polarization of the Jacobian in Torelli's theorem
Current License: CC BY-SA 4.0
9 events
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| Nov 13, 2018 at 8:21 | comment | added | Xarles | About the second question, may be it is possible to use that the Riemann period matrix of the product of two elliptic curves is diagonal, while for a genus 2 curves is not (but I am not sure how to show this "elementarily"). | |
| Nov 13, 2018 at 2:13 | history | edited | Manoel |
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| Nov 13, 2018 at 0:28 | comment | added | roy smith | As abx nicely observes, that h^0(D) = 1 (and that h^j(D) = 0 for j>0), are seen most clearly and directly here by the Kunneth formula. But if you are curious how someone knew immediately that D (and C) defines a principal polarization, they may also have mentally calculated an intersection number, since an effective divisor on an abelian variety of dimension g is a principal polarization iff its self intersection number is g! This can be seen immediately for D, (and for C by the adjunction formula). The cohomology calculations follow in general, but for D Kunneth just lays them out. | |
| Nov 12, 2018 at 6:00 | review | Close votes | |||
| Nov 13, 2018 at 22:16 | |||||
| Nov 12, 2018 at 5:40 | comment | added | abx | For the first question, it is more direct to use the Künneth formula. | |
| Nov 12, 2018 at 2:06 | comment | added | roy smith | forgive me if not precise, but just off the top of my head, first question: use riemann roch (i.e.compute intersection numbers); second question: the theta divisors are not isomorphic. | |
| Nov 12, 2018 at 1:13 | history | edited | Manoel | CC BY-SA 4.0 |
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| Nov 11, 2018 at 23:45 | history | edited | Manoel | CC BY-SA 4.0 |
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| Nov 11, 2018 at 23:34 | history | asked | Manoel | CC BY-SA 4.0 |