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Let $k=\mathbb{C}$. Call a smooth projective surface $X$ an anti-canonical divisor if there is a smooth projective $3$-fold $Y$ with a section $s \in H^{0}(Y,-K_{Y})$ such that $\{s=0\} \cong X$.

Question: Are anti-canonical divisors classified?

I asked this on math stack exchange 7 months ago with no answer (https://math.stackexchange.com/questions/3311295/smooth-surfaces-appearing-as-an-anticanonical-section) so I ask it here.

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    $\begingroup$ 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? $\endgroup$
    – abx
    Commented Mar 3, 2020 at 12:39
  • $\begingroup$ I mean classification up to isomorphism. $\endgroup$
    – Nick L
    Commented Mar 3, 2020 at 12:52
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    $\begingroup$ Is "K3 or abelian" a satisfactory answer? If not, what kind of description do you expect? $\endgroup$
    – abx
    Commented Mar 3, 2020 at 13:51
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    $\begingroup$ 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. $\endgroup$
    – Tony Pantev
    Commented Mar 3, 2020 at 14:07
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    $\begingroup$ 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 $\endgroup$
    – Evgeny Shinder
    Commented Mar 20, 2020 at 23:05

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