  We investigate in ZFC what can be the family of large enough cardinals mu in
which an a.e.c. K is categorical or even just solvable. We show that for not
few cardinals lambda<mu there is a superlimit model in K_lambda. Moreover, our
main result is that we can find a good lambda-frame s categorical in lambda
such that K_s subseteq K_lambda. We then show how to use [Sh:705] to get
categoricity in every large enough cardinality if K has cases of
mu-amalgamation for enough mu and 2^mu<2^{mu^{+1}} <... < 2^{mu^{+n}}... for
enough mu.
