  We describe the equations and Gr\"obner bases of some degenerate K3 surfaces
associated to rational normal scrolls. These K3 surfaces are members of a class
of interesting singular projective varieties we call correspondence scrolls.
The ideals of these surfaces are nested in a simple way that allows us to
analyze them inductively. We describe explicit Gr\"obner bases and syzygies for
these objects over the integers and this lets us treat them in all
characteristics simultaneously.
