It has been suggested that a $ u$p process can occur when hot, dense, and proton-rich matter is expanding within a strong flux of anti-neutrinos. In such an environment, proton-rich nuclides can be produced in sequences of proton captures and (n,p) reactions, where the free neutrons are created in situ by $overline{ u}_mathrm{e}+mathrm{p} rightarrow mathrm{n}+mathrm{e}^+$ reactions. The detailed hydrodynamic evolution determines where the nucleosynthesis path turns off from N = Z line and how far up the nuclear chart it runs. In this work, the uncertainties on the final isotopic abundances stemming from uncertainties in the nuclear reaction rates were investigated in a large-scale Monte Carlo approach, simultaneously varying ten thousand reactions. A large range of model conditions was investigated because a definitive astrophysical site for the $ u$p process has not yet been identified. The present parameter study provides, for each model, identification of the key nuclear reactions dominating the uncertainty for a given nuclide abundance. As all rates appearing in the $ u$p process involve unstable nuclei, and thus only theoretical rates are available, the final abundance uncertainties are larger than those for nucleosynthesis processes closer to stability. Nevertheless, most uncertainties remain below a factor of three in trajectories with robust nucleosynthesis. More extreme conditions allow production of heavier nuclides but show larger uncertainties because of the accumulation of the uncertainties in many rates and because the termination of nucleosynthesis is not at equilibrium conditions. It is also found that the solar ratio of the abundances of ${}^{92}$Mo and ${}^{94}$Mo could be reproduced within uncertainties.
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