Moduli spaces of quartic hyperelliptic K3 surfaces via K-stability

A general polarized hyperelliptic K3 surfaces of degree 4 is a double cover of P^1 x P^1  branched along a bidegree (4,4) curve. Classically there are two compactifications of their moduli spaces: one is the GIT quotient of (4,4) curves, the other is the Baily-Borel compactification of their periods. We show that K-stability provides a natural modular interpolation between these two compactifications. This provides a new aspect toward a recent result of Laza-O'Grady. Based on joint work in progress with K. Ascher and K. DeVleming.