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(short version) - A model for GRMHD disk outflows with neutrino-driven massejection is developed,and employed to calculate the structure of the outflow inthe sub-slow magnetosonic region and the mass loading of the outflow, underconditions anticipated in the central engines of gamma-ray bursts. Thedependence of the mass flux on the conditions in the disk, on magnetic fieldgeometry, and on other factors is carefully examined for a range of neutrinoluminosities expected in hyperaccreting black holes. The fraction of neutrinoluminosity that is being converted to kinetic energy flux is shown to be asensitive function of the effective neutrino temperature at the flow injectionpoint, and the shape of magnetic field lines in the sub-slow region, but ispractically independent of the strength of poloidal and toroidal magneticfields. We conclude that magnetic launching of ultra-relativistic polaroutflows from the innermost parts of the disk is in principle possible providedthe neutrino luminosity is sufficiently low, $L_\nu\simlt10^{52}$ erg s$^{-1}$or so. The conditions found to be optimal for the launching of anultra-relativistic jet are also the conditions favorable for largeneutron-to-proton ratio in the disk.Comment: 28 pages, 8 figures, ApJ in press. Post refereed version, more  discussion plus additional figure adde

General Relativistic, Neutrino‐assisted Magnetohydrodynamic Winds—Theory and Application to Gamma‐Ray Bursts. I. Schwarzschild Geometry