The Kodaira classification of the moduli of hyperelliptic curves

We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces are much worst from the point of view of singularities than the moduli space of pointed stable curves M_{g,n}, but much better from the point of view of birational complexity. We provide a full Kadaira classification with the surprising feature that for n large fixed, the Kodaira dimension decreases as the genus grows. In other words, the space becomes "simpler" as the genus grows. Further, we study the polyhedrality of the effective cone. This is joint work with Scott Mullane.