The moduli space of polarized K3 surfaces of a given degree is identified with a locally symmetric variety, and hence has a Baily-Borel compactification. On the other hand, GIT quotients of m-embedded polarized K3’s provide different (projective) birational models of the Baily-Borel compactification. E. Looijenga has constructed a framework that allows to compare these compactifications. In this talk, I will discuss an enrichment of this picture, essentially a continuous interpolation between the GIT and BB models. While the discussion will be mostly concerned with the case of hyperelliptic quartic K3 surfaces, we expect such an interpolation to hold quite generally. There is a strong analogy with the Hassett-Keel program that studies the birational geometry of the moduli space of curves.
This is a report on joint work with K. O’Grady.