Abstract

Recently, there has been significant progress in understanding the K-moduli spaces of Fano varieties and log Fano pairs (X,cD). When D is a rational multiple of the anticanonical divisor of X, the K-moduli spaces of log Fano pairs (X,cD) admit a wall crossing framework as c varies and there is a finite collection of rational values of c where the K-moduli spaces change. With Lena Ji, Patrick Kennedy-Hunt, and Ming Hao Quek, we explore the K-moduli spaces in an example where D is not proportional to the anticanonical divisor. We study the K-moduli space of pairs (P1xP2, cD) where D is a (2,2) divisor and prove that there is exactly one irrational value of c where the moduli spaces change. We further relate these moduli spaces to several related spaces: the GIT of (2,2) divisors in P1xP2, K-moduli of the conic bundle threefold that is the double cover of P1xP2 branched over D, and various moduli spaces of quartic plane curves arising as the discriminant of these conic bundles.