Marangoni convection in a cavity subject to point (concentrated) heating has been investigated. The analysis includes the complete effects of the interface deformation. The results determined for large Biot and zero Marangoni (zero Prandtl) numbers show that steady convection may exist only for a limited range of Reynolds numbers Re (bounded from above and from below), and for capillary numbers Ca and cavity lengths L smaller than certain critical values. The main factor limiting the existence of steady convection involves the interface approaching the bottom of the cavity. Unsteady analysis shows that when the conditions guaranteeing the existence of steady convection are not met, an interface rupture process sets in leading, eventually, to the formation of a dryout at the bottom of the cavity. The initial stages of the rupture process are characterized by a rapidly accelerating growth of the interface deformation. The critical values of Re, Ca and L, which guarantee the existence of steady convection, are mutually dependent and change with the heating rate; they reach a minimum for instantaneous heating. Too rapid heating produces an oscillatory transient which always decays in the range of parameters studied.