  We discuss fattening phenomenon for the evolution of planar curves according
to their nonlocal curvature. More precisely, we consider a class of generalized
curvatures which correspond to the first variation of suitable nonlocal
perimeter functionals, defined in terms of an interaction kernel $K$, which is
symmetric, nonnegative, possibly singular at the origin, and satisfies
appropriate integrability conditions.
  We prove a general result about uniqueness of the geometric evolutions
starting from regular sets with positive $K$-curvature and we discuss the
fattening phenomenon for the evolution starting from the cross, showing that
this phenomenon is very sensitive to the strength of the interactions. As a
matter of fact, we show that the fattening of the cross occurs for kernels with
sufficiently large mass near the origin, while for kernels that are
sufficiently weak near the origin such a fattening phenomenon does not occur.
  We also provide some further results in the case of the fractional mean
curvature flow, showing that strictly starshaped sets have a unique geometric
evolution.
  Moreover, we exhibit two illustrative examples of closed nonregular curves,
the first with a Lipschitz-type singularity and the second with a cusp-type
singularity, given by two tangent circles of equal radius, whose evolution
develops fattening in the first case, and is uniquely defined in the second,
thus remarking the high sensitivity of the fattening phenomenon in terms of the
regularity of the initial datum. The latter example is in striking contrast to
the classical case of the (local) curvature flow, where two tangent circles
always develop fattening.
  As a byproduct of our analysis, we provide also a simple proof of the fact
that the cross is not a $K$-minimal set for the nonlocal perimeter functional
associated to $K$.
