  We consider a core-radius approach to nonlocal perimeters governed by
isotropic kernels having critical and supercritical exponents, extending the
nowadays classical notion of $s$-fractional perimeter, defined for $0<s<1$, to
the case $s\ge 1$\,.
  We show that, as the core-radius vanishes, such core-radius regularized
$s$-fractional perimeters, suitably scaled, $\Gamma$-converge to the standard
Euclidean perimeter. Under the same scaling, the first variation of such
nonlocal perimeters gives back regularized $s$-fractional curvatures which, as
the core radius vanishes, converge to the standard mean curvature; as a
consequence, we show that the level set solutions to the corresponding nonlocal
geometric flows, suitably reparametrized in time, converge to the standard mean
curvature flow.
  Finally, we prove analogous results in the case of anisotropic kernels with
applications to dislocation dynamics. Keywords: Fractional perimeters;
$\Gamma$-convergence; Local and nonlocal geometric evolutions; Viscosity
solutions; Level set formulation; Fractional mean curvature flow; Dislocation
dynamics
