A physically based parameterization for the nucleation and initial growth of ice crystals in cirrus clouds at low (<238 K) temperatures prevailing in the upper troposphere and in the tropopause region is implemented in the ECHAM general circulation model (GCM). With this model we performed the first interactive simulations of cirrus clouds in a global climate model. As homogeneous ice crystal nucleation requires relative humidities with respect to ice of 140--170% at these temperatures, we abandoned the saturation adjustment scheme previously used to compute the properties of cirrus clouds in GCMs. The comparison with relative humidity data taken on commercial aircraft showed that ECHAM reproduces the frequency distribution of supersaturation with respect to ice in cloud-free regions well especially if high values of the subgrid-scale vertical velocity are assumed. On the other hand, the higher frequency of occurrence of high supersaturations in the model may point to heterogeneous freezing as the missing mechanism for ice formation. We found that homogeneous freezing may be limited by the number of hygroscopic aerosols present. However, as the ice water content remains the same for the natural and present-day aerosol scenarios the longwave radiation does not change. Thus anthropogenic aerosol and precursor emissions have only a small effect on cirrus formed by homogeneous freezing. Aircraft emissions of sulfate aerosols are not likely to be important for cirrus formation. Aircraft soot emissions may be important if the soot particles nucleate ice more efficiently than by homogeneous freezing.