Fano–K3 reconstruction and isomorphism to the K3 moduli

Establish that for every ADE K3 surface (S,L) in the degree-22 K3 moduli stack F22 there exists a unique Gorenstein canonical Fano threefold X admitting a Q-Gorenstein smoothing to a smooth V22 such that S lies in the anticanonical linear system |-K_X| and L equals -K_X restricted to S; moreover, show that the forgetful map from the moduli stack of plt pairs P^{plt} parametrizing such (X,S) to F22 is an isomorphism.

Background

The paper proves that for K-semistable degenerations of V22 the anticanonical K3 surface controls the moduli and shows an open immersion from a pair-moduli stack to the K3 moduli. This suggests a stronger reconstruction principle: the ambient Fano threefold should be uniquely recoverable from its anticanonical K3 surface.

The conjecture formulates this expected bijective correspondence globally, asserting both existence and uniqueness of the Fano threefold X for every polarized ADE K3 surface (S,L) of degree 22 and the isomorphism of the forgetful morphism of moduli stacks.

References

Conjecture For each ADE K3 surface $(S,L)\in F_{22}$, there exists a unique Gorenstein canonical Fano threefold $X$ admitting a $Q$-Gorenstein smoothing to $V_{22}$ such that $S \in |-K_X|$ and $L = -K_X|S$. Moreover, the forgetful map $P{plt} \longrightarrow F{22}, \qquad (X,S) \longmapsto (S, -K_X|_S),$ is an isomorphism.

— The boundary of K-moduli of prime Fano threefolds of genus twelve  (2603.29827 - Kaloghiros et al., 31 Mar 2026) in Section 5 (Reconstruction of Fano threefolds from K3 surfaces), Conjecture 5.1