  In the first part of this paper, we investigate the sharp threshold of
blow-up and global existence for the focusing nonlinear Schr\"{o}dinger
equation with combined nonlinearities of mass-critical and mass-subcritical
power-type. Especially, we find a sequence of initial data with mass
approximating that of the ground state from above, the correspondng solution of
which blows up. This result partially gives a positive answer to the open
problem left by T.Tao, M. Visan and X. Zhang in \cite{TVZ}. After then, we
obtain the lower blow-up rate and concentration rate for the blow-up solutions
in the same case. Finally, we consider the mass-critical combined with the
mass-supercritical power type case, studying the blow-up criteria, blow-up rate
and concentration of the blow-up solutions in this case.
