Timeline for Nef classes on abelian varieties in positive characteristic
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 21, 2013 at 5:59 | comment | added | naf | I don't see why you need this condition. It suffices to descend to $A/B$ where $B$ is the identity component of $K(L)$; the pullback of the theta group to $B$ is then commutative and what you need to show is that the corresponding extension splits. Assuming $L$ symmetric allows one to define a symmetric isomorphism $T_x^*L \cong L$. By this I mean that this isomorphism is preserved by $(-1)^*$ (after using the fixed identification of $L$ with $(-1)^*L$ and translating by $x$). This is well defined after squaring and should give the required splitting. | |
| Jul 18, 2013 at 20:53 | comment | added | rfauffar | After reading a bit, I see that one of the conditions necessary for $L$ to descend is that the commutator map $e^{L^2}:K(L^2)\times K(L^2)\to k^\times$ be trivial on $K(L)\times K(L)$. I don't see why this should happen by only asking for $L$ to be symmetric... | |
| Jul 12, 2013 at 15:11 | comment | added | rfauffar | Ok great, I'm going to take a look at that book. Thanks a lot! | |
| Jul 12, 2013 at 13:11 | comment | added | naf | This is a special case of flat descent; see, for example, Theorem 4 in Chapter 6 of the book Neron Models by Bosch, Lutkebohmert and Raynaud. (The situation here is simpler though since the morphism $A \to A/B$ is a smooth homomorphism so you should be able to rewrite the condition for the existence of descent datum). After fixing an isomorphism of $L$ with $(-1)^*L$, there is a symmetric isomorphism $T_x^*L \cong L$ which is well defined up to sign. The sign ambiguity disappears if you square these isomorphisms. | |
| Jul 11, 2013 at 21:16 | comment | added | rfauffar | Where can I read about descent, in this sense? Why does the data you gave assure me that $L^2$ descends to a line bundle on $A/B$? | |
| Jul 11, 2013 at 19:42 | vote | accept | rfauffar | ||
| Jul 7, 2013 at 6:01 | history | made wiki | Post Made Community Wiki by naf | ||
| Jul 7, 2013 at 5:48 | history | answered | naf | CC BY-SA 3.0 |