  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.
