The $ u p$ process appears in proton-rich, hot matter which is expanding in a neutrino wind and may be realised in explosive environments such as core-collapse supernovae or in outflows from accretion disks. The impact of uncertainties in nuclear reaction cross sections on the finally produced abundances has been studied by applying Monte Carlo variation of all astrophysical reaction rates in a large reaction network. As the detailed astrophysical conditions of the $ u p$ process still are unknown, a parameter study was performed, with 23 trajectories covering a large range of entropies and $Y_mathrm{e}$. The resulting abundance uncertainties are given for each trajectory. The $ u p$ process has been speculated to contribute to the light $p$ nuclides but it was not possible so far to reproduce the solar isotope ratios. It is found that it is possible to reproduce the solar $^{92}$Mo/$^{94}$Mo abundance ratio within nuclear uncertainties, even within a single trajectory. The solar values of the abundances in the Kr-Sr region relative to the Mo region, however, cannot be achieved within a single trajectory. They may still be obtained from a weighted superposition of different trajectories, though, depending on the actual conditions in the production site. For a stronger constraint of the required conditions, it would be necessary to reduce the uncertainties in the 3$alpha$ and $^{56}$Ni(n,p)$^{56}$Co rates at temperatures $T>3$ GK.
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