Timeline for anticanonical divisors
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 20, 2020 at 23:05 | comment | added | Evgeny Shinder | If you also require that $-K_Y$ is ample, so that $Y$ is a Fano threefold, then say in Picard rank one, K3 surface $X$ will be of degree 2, 4, 6, 8, 10, 12, 14, 16, 18, 22, which is part of the classification of Fano threefolds. Beauville generalizes this for arbitrary Picard rank (keeping the Fano assumption): arxiv.org/pdf/math/0211313.pdf | |
| Mar 3, 2020 at 14:07 | comment | added | Tony Pantev | It seems clear that every abelian surface $X$ can be realized as an anti-canonical divisor since we can view $X$ as an etale double cover of another abelian surface $Y$ and so $X$ embeds as an anti-canonical divisor in the threefold $\mathbb{P}(\mathcal{O}_{Y}\oplus L)$ where $L \to Y$ is the two torsion line bundle corresponding to the cover $X \to Y$. So the OP seems to be asking - which smooth K3 surfaces can be realized as anti-canonical divisors of smooth threefolds. I am not sure where this is going since obviously every K3 is a connected component of an anti-canonical divisor. | |
| Mar 3, 2020 at 13:51 | comment | added | abx | Is "K3 or abelian" a satisfactory answer? If not, what kind of description do you expect? | |
| Mar 3, 2020 at 12:52 | comment | added | Nick L | I mean classification up to isomorphism. | |
| Mar 3, 2020 at 12:39 | comment | added | abx | What do you mean by "classified"? By the adjunction formula your surface has trivial canonical bundle, hence is a K3 or an abelian surface, and both cases can occur. What else? | |
| Mar 3, 2020 at 11:18 | history | edited | Nick L | CC BY-SA 4.0 |
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| Mar 3, 2020 at 10:42 | history | asked | Nick L | CC BY-SA 4.0 |