  We analyze event-by-event fluctuations of the transverse momentum in
relativistic heavy-ion collisions at RHIC in the framework based on the
fluctuating Glauber-model initial conditions, event-by-event (3+1)-dimensional
viscous hydrodynamics, and statistical hadronization. We use the scaled
fluctuation measure < Delta p_Ti Delta p_Tj > / << p_T >>. The identified
"geometric" mechanism of generating the transverse-momentum fluctuations from
the initial size fluctuations, transmitted to the final
statistical-hadronization phase with hydrodynamics, is capable of easily
reproducing the magnitude of the effect and explains the basic features of the
data. On the other hand, it is somewhat too strong, hinting on modification of
the popular Glauber approach to the earliest phase of the collision. We have
checked that the considered measure is insensitive of the values of the shear
and bulk viscosity coefficients, the freeze-out temperature, and the smoothing
parameter for the initial distribution. It remains unaltered in the core-corona
picture and is insensitive to the transverse-momentum conservation,
approximately imposed in the statistical hadronization.
