On the Chow Rings of the Moduli Spaces ${\mathcal{M}}_{5,8}$ and ${\mathcal{M}}_{5,9}$

In this paper we prove that the rational Chow rings of ${\mathcal{M}}_{5,8}$ and ${\mathcal{M}}_{5,9}$ are tautological, and that these moduli spaces have the Chow Künneth generation Property.