Q-Gorenstein smoothability for seven Noether–Lefschetz divisors

Ascertain whether, for each of the seven specified Noether–Lefschetz divisors in the degree-22 K3 moduli, the Gorenstein canonical Fano threefolds constructed by the authors that contain a general member K3 surface as an anticanonical divisor have Picard rank 1 and admit a Q-Gorenstein smoothing to a smooth V22 threefold.

Background

The authors construct, for each of seven Noether–Lefschetz divisors complementary to the image of the open immersion, explicit Gorenstein canonical Fano threefolds containing the given K3 surfaces as anticanonical divisors. These serve as candidates to extend the Fano–K3 correspondence beyond the K-semistable locus.

While partial verifications are given for several cases, the general Picard rank and smoothability to V22 are not established; the conjecture encapsulates this remaining step to align these degenerations with the V22 family.