GIT versus Baily-Borel compactification for $K3$'s which are double covers of $\mathbb P^1\times\mathbb P^1$

In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper (arXiv:1607.01324), based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper (arXiv:1612.07432) we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of our conjectures for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for (2,4) complete intersection curves.

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