The Chow ring of the moduli space of degree $2$ quasi-polarized K3 surfaces

We study the Chow ring with rational coefficients of the moduli space ${\mathcal{F}}_2$ of quasi-polarized $K3$ surfaces of degree $2$. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group is ${\mathsf{A}}^{17}({\mathcal{F}}_2)\cong \mathbb{Q}$. We prove that the Chow ring consists of tautological classes and is isomorphic to the even cohomology. The Chow ring is not generated by divisors and does not satisfy duality with respect to the pairing into ${\mathsf{A}}^{17}({\mathcal{F}}_2)$. The kernel of the pairing is a 1-dimensional subspace of ${\mathsf{A}}^9({\mathcal{F}}_2)$ which we calculate explicitly. In the appendix, we revisit Kirwan-Lee’s calculation of the Poincaré polynomial of ${\mathcal{F}}_2$.