The optimal L^4-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus T^2 is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger L4 bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in H^s(T^2) for any s>0 and data which is small in the critical norm