Equations and Syzygies of K3 Carpets and Unions of Scrolls

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.