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By work of Looijenga and others, one has a good understanding of therelationship between GIT and Baily-Borel compactifications for the modulispaces of degree 2 K3 surfaces, cubic fourfolds, and a few other relatedexamples. The similar-looking cases of degree 4 K3 surfaces and double EPWsextics turn out to be much more complicated for arithmetic reasons. In thispaper, we refine work of Looijenga to allow us to handle these cases.Specifically, in analogy with the so-called Hassett-Keel program for the modulispace of curves, we study the variation of log canonical models for locallysymmetric varieties of Type IV associated to D lattices. In particular, for thedimension 19 case, we conjecturally obtain a continuous one-parameterinterpolation (via a directed MMP) between the GIT and Baily-Borelcompactifications for the moduli of degree 4 K3 surfaces. The predictions forthe similar 18 dimensional case, corresponding geometrically to hyperellipticdegree 4 K3 surfaces, can be verified by means of VGIT -- this will bediscussed elsewhere.

Birational geometry of the moduli space of quartic K3 surfaces