  We consider the nonlinear Schr\"odinger equation with periodic dispersion
management. We first establish global-in-time Strichartz estimates for the
underlying linear equation with suitable dispersion maps. As an application, we
establish a small-data scattering result for the $3d$ cubic equation. Finally,
we use a virial argument to demonstrate the existence of blowup solutions for
the $3d$ cubic equation with piecewise constant dispersion map.
