The K-moduli space of a family of conic bundle threefolds

We describe the 6-dimensional compact K-moduli space of Fano threefolds in deformation family No 2. 18. These Fano threefolds are double covers of P¹ P² branched along smooth (2, 2) -surfaces, and Cheltsov--Fujita--Kishimoto--Park proved that any smooth Fano threefold in this family is K-stable. A member of family No 2. 18 admits the structures of a conic bundle and a quadric surface bundle. We prove that K-polystable limits of these Fano threefolds admit conic bundle structures, but not necessarily del Pezzo fibration structures. We study this K-moduli space via the moduli space of log Fano pairs (P¹ P², c R) for c=1/2 and R a (2, 2) -divisor, which we construct using wall-crossings. In the case where the divisor is proportional to the anti-canonical divisor, the first author, together with Ascher and Liu, developed a framework for wall crossings in K-moduli and proved that there are only finitely many walls, which occur at rational values of the coefficient c. This paper constructs the first example of wall-crossing in K-moduli in the non-proportional setting, and we find a wall at an irrational value of c. In particular, we obtain explicit descriptions of the GIT and K-moduli spaces (for c 1/2) of these (2, 2) -divisors. Furthermore, using the conic bundle structure, we study the relationship with the GIT moduli space of plane quartic curves.