We derive the asymptotical dynamical law for Ginzburg-Landau
vortices in the plane under the Schrödinger dynamics, as the
Ginz\-burg-Landau parameter goes to zero. The limiting law is the
well-known point-vortex system. This result extends to the whole
plane previous results of [Colliander, J.E. and Jerrard, R.L.:
Vortex dynamics for the Ginzburg-Landau-Schrödinger equation.
Internat. Math. Res. Notices 1998, no. 7, 333-358; Lin, F.-H. and Xin, J.\,X.:
On the incompressible fluid limit and the vortex motion law of the nonlinear
Schr\"{o}dinger equation. Comm. Math. Phys. 200 (1999), 249-274] established for bounded
domains, and holds for arbitrary degree at infinity. When this
degree is non-zero, the total Ginzburg-Landau energy is infinite.