The moduli space of genus $8$ Fano $3$-folds with two $\frac{1}{2}(1,1,1)$-points: rationality and K-stability

Abstract

Let ${\mathcal{F}}_{8,2\times \frac{1}{2}(1,1,1)}$ be the moduli space of genus $8$ Fano $3$-folds with two terminal quotient singularities of type $\frac{1}{2}(1,1,1)$. We prove that ${\mathcal{F}}_{8,2\times \frac{1}{2}(1,1,1)}$ is rational. Moreover, we show that a general member of this family is K-stable.

37 pages