Timeline for Uniqueness of theta divisor
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 6, 2020 at 22:12 | comment | added | TartagliaTriangle | Thanks @Hacon for the useful comment! I'll check it out | |
| Mar 6, 2020 at 20:03 | comment | added | Hacon | By M.S. NARASIMHAN AND M. V. NORI, Polarizations on an abelian variety, Proc. Indian Accad. Sci. Math. Sci. 90 (1981) No. 2, 125-128. any abelian variety admits only a finite number of principal polarizations up to isomorphism. To compute the number of these isomorphism classe, see "Abelian varieties with several principal polarizations" by Herbert Lange available at projecteuclid.org/euclid.dmj/1077306167 | |
| Mar 5, 2020 at 16:17 | comment | added | TartagliaTriangle | @JoeSilverman By theta divisor I mean a divisor whose associated ample line bundle gives a principal polarization. I'm not interested in Jacobian varieties. | |
| Mar 5, 2020 at 15:08 | comment | added | Joe Silverman | I think maybe the terminology is confusing. Are you using theta divisor as a synonym for a divisor that gives a principal polarization on an arbitrary abelian variety? Then you'll find lots of counterexamples that are not algebraically equivalent. Or are you using it to be a translate of the image of $C^{g-1}$ in its Jacobian variety? | |
| Mar 5, 2020 at 14:55 | history | edited | TartagliaTriangle | CC BY-SA 4.0 |
The question was edited following comments.
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| Mar 5, 2020 at 5:21 | comment | added | abx | @Felipe Voloch: Indeed I probably misunderstood the question, which is far from clear. | |
| Mar 5, 2020 at 0:47 | comment | added | Felipe Voloch | I think @abx misunderstood your question. As I said, the answer is no. | |
| Mar 4, 2020 at 21:17 | comment | added | TartagliaTriangle | @FelipeVoloch Yes, my question is: if I have two theta divisor on $A$, do they end up being the same (up to something)? | |
| Mar 4, 2020 at 18:44 | comment | added | Felipe Voloch | Are you asking whether an abelian variety has a unique principal polarization, if one exists? The answer is no. You'll find examples by searching for non-isomorphic curves with isomorphic jacobians, for instance. | |
| Mar 4, 2020 at 16:50 | review | Close votes | |||
| Mar 9, 2020 at 3:05 | |||||
| Mar 4, 2020 at 16:29 | comment | added | abx | The line bundle satisfies $h^0(\mathscr{O}_A(\Theta ))=1$, hence it corresponds to a unique effective divisor. And a polarization defines the corresponding line bundle up to translation. | |
| Mar 4, 2020 at 16:03 | comment | added | TartagliaTriangle | @abx I'm sorry I don't get it! For me a theta divisor is a divisor whose associated line bundle gives a principal polarization. From that, how can I prove the uniqueness? | |
| Mar 4, 2020 at 15:54 | comment | added | abx | Indeed, the theta divisor is unique up to translation, essentially by definition. See any book on abelian varieties, for instance Birkenhake-Lange. | |
| Mar 4, 2020 at 15:09 | history | asked | TartagliaTriangle | CC BY-SA 4.0 |