Timeline for Examples such that Albanese morphism is not well-defined
Current License: CC BY-SA 3.0
8 events
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| Jan 1, 2018 at 8:24 | comment | added | 1984 | @JasonStarr , I think you was right, these two notion are the same in characteristic zero for normal projective varieties with only rational singularities , in fact, Serre introduced it for quas-Albanese morphism , but I don't have example to have good picture on both of them. See Lemma 8.1 of Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. 2 | |
| Dec 31, 2017 at 22:30 | comment | added | 1984 | @JasonStarr, Thank you, interesting comment. I am interested in Albanese morphism | |
| Dec 31, 2017 at 22:28 | comment | added | 1984 | @AriyanJavanpeykar, thank you for the interesting comments. However you didn't mention your example | |
| Dec 31, 2017 at 22:25 | history | edited | 1984 | CC BY-SA 3.0 |
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| Dec 31, 2017 at 15:32 | comment | added | Jason Starr | What @AriyanJavanpeykar writes is true. However, my guess is that the OP is not asking about the Albanese morphism, but rather the "birational Albanese morphism", i.e., for a specified smooth $\mathbb{C}$-point $x_0$, an initial object in the category of rational transformations $f:X\dashrightarrow A$ to Abelian varieties that are regular at $x_0$ and that map $x_0$ to the origin. In this case there are many birational Albanese morphisms that are not regular, e.g., when $X$ is a projective cone over a smooth plane cubic. | |
| Dec 31, 2017 at 11:01 | comment | added | Ariyan Javanpeykar | Proposition A.6 of kurims.kyoto-u.ac.jp/~motizuki/… on p. 72 says that any proper normal integral scheme $X$ over a perfect field $k$ with a $k$-rational point admits an Albanese morphism $X\to Alb(X)$. If you allow non-normal singularities, then you can probably find (very singular) curves which do not have an Albanese variety. | |
| Dec 31, 2017 at 10:06 | history | edited | 1984 | CC BY-SA 3.0 |
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| Dec 31, 2017 at 8:26 | history | asked | 1984 | CC BY-SA 3.0 |