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By K3 surfaces and Fano threefolds, I mean smooth ones.

If a K3 surface $S$ is an anticanonical section of a Fano threefold $V$ of Picard rank one (hence, $Pic(V)=\mathbb Z H_V$ for some ample divisor $H_V$), then $h:=H_V|_S$ is a primitive ample divisor of $S$. A divisor class $h$ is said to be primitive if $h=nD$ for divisor class $D$ with $n\in \mathbb Z$ implies $n=\pm 1$.

According to the classification of Fano threefolds, $4 \le h^2 \le 22$.

My question is some converse of it:

Let $S$ be a K3 surface with very ample primitive divisor $h$ ( $4 \le h^2 \le 22$). Then can $S$ be embedded into a Fano threefold $V$ of Picard rank one such that $h=H_V|_S$?

If $h^2=4, 6, 8$, the linear system $|h|$ embeds $S$ into projective spaces as a quartic surface, a complete intersection of quadric and cubic hypersurfaces and a complete intersection of three quadric hypersurfaces respectively. So the answer to the above question is yes in these cases.

But I am not sure about the cases of $h^2=10, 12, ..., 22$. Does anyone know the answer for any cases of those?

By the way, the answer is yes for generic K3 surfaces ([https://arxiv.org/abs/math/0211313]) but I am wondering whether one can remove the condition `generic'.

Improve this question
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  • $\begingroup$ For the locally free sheaf of principal parts on $V$, $\mathcal{P}^1_V(\omega_V)$, that occurs in the definition of the Atiyah extension, you can try to prove vanishing of $H^1(S,\textit{Hom}_{\mathcal{O}_V}(\mathcal{P}^1_V(\omega_V),\omega_V)|_S)$. This would follow from vanishing of both $H^1(V,\textit{Hom}_{\mathcal{O}_V}(\mathcal{P}^1_V(\omega_V),\omega_V))$ and $H^2(V,\textit{Hom}_{\mathcal{O}_V}(\mathcal{P}^1_V(\omega_V),\omega_V)\otimes \omega_V)$. $\endgroup$
    – Jason Starr
    Commented Jul 8, 2022 at 1:05
  • $\begingroup$ To continue the previous comment, there is a short exact sequence by "contracting" with the class of the anticanonical section: $0\to H^{1,2}(V)\to H^1(V,T_V) \to H^1(S,\textit{Hom}_{\mathcal{O}_V}(\mathcal{P}^1(\omega_V),\omega_V)|_S) \to 0.$ So whenever the first map is surjective, then deformations of the polarized K3 arise from deformations as embedded anticanonical divisors in $V$. Do you want to allow to deform $V$ as well as the divisor $S$ in the anticanonical linear system on $V$? $\endgroup$
    – Jason Starr
    Commented Jul 8, 2022 at 11:21
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    $\begingroup$ Now I understand what the OP is asking. The computation in my comments shows that there is an open locus in the moduli stack of polarized K3 surfaces parameterizing those that embed as anticanonical divisors in some Fano threefold $V$. I believe the OP is asking whether this open substack equals the entire moduli stack. The answer is almost certainly "no" unless the Fano threefold $V$ is allowed to be singular or degenerate in some other way (e.g., allow the anticanonical divisor class to be $\mathbb{Q}$-big and nef). $\endgroup$
    – Jason Starr
    Commented Jul 8, 2022 at 11:28
  • $\begingroup$ It looks to me like $h^2=12$ gives counterexamples: some of these K3 surfaces are anticanonical divisors in the weighted projective threefold $V=\mathbb{P}(1,1,1,3)$. $\endgroup$
    – Jason Starr
    Commented Jul 8, 2022 at 16:16
  • $\begingroup$ There are smooth Fano threefolds with $H_V^3=12$ with index one. Do you mean that the K3 surface in $P(1,1,1,3)$ cannot be embedded into any of those Fano threefolds? $\endgroup$
    – Basics
    Commented Jul 8, 2022 at 18:11

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