  We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It
is known that, if the Kawasaki part is speeded up in a diffusive space-time
scaling, one can derive the Allen-Cahn equation which is a kind of the
reaction-diffusion equation in the limit. This paper concerns the scaling that
the Glauber part, which governs the creation and annihilation of particles, is
also speeded up but slower than the Kawasaki part. Under such scaling, we
derive directly from the particle system the motion by mean curvature for the
interfaces separating sparse and dense regions of particles as a combination of
the hydrodynamic and sharp interface limits.
