  In recent experiments with a superconducting nanosized Pb bridge formed
between a scanning tunneling microscope tip and a substrate, superconductivity
has been detected at magnetic fields, which are few times larger than the third
(surface) critical field. We describe the observed phenomenon on the basis of a
numerical solution of the Ginzburg-Landau equations in a model structure
consisting of six conoids. The spatial distribution of the superconducting
phase is shown to be strongly inhomogeneous, with concentration of the
superconducting phase near the narrowest part (the ``neck'') of the bridge. We
show that suppression of superconductivity in the bridge by applied magnetic
field or by temperature first occurs near the bases and then in the neck
region, what leads to a continuous superconducting-to-normal resistive
transition. A position of the transition midpoint depends on temperature and,
typically, is by one order of magnitude higher than the second critical field
H_c2. We find that the vortex states can be realized in the bridge at low
temperatures T/T_c < 0.6. The vortex states lead to a fine structure of the
superconducting-to-normal resistive transition. We also analyze vortex states
in the bridge, which are characterized by a varying vorticity as a function of
the bridge's height.
