  A ccc-generically supercompact cardinal $\kappa$ can be smaller than or equal
to the continuum. On the other hand, such a cardinal $\kappa$ still satisfies
diverse largeness properties, like that it is a stationary limit of
ccc-generically measurable cardinals (Theorem 4.1). This is in a strong
contrast to $\cal P$-generically supercompact cardinals for the class $\cal P$
of all $\sigma$-closed posets, which can be $\aleph_n$ for any n>1.
