We present nucleosynthesis calculations and the resulting ¹⁹F stellar yields for a large set of models with different masses and metallicity. During the asymptotic giant branch (AGB) phase, ¹⁹F is produced as a consequence of nucleosynthesis occurring during the convective thermal pulses and also during the interpulse periods if protons from the envelope are partially mixed in the top layers of the He intershell (partial mixing zone). We find that the production of fluorine depends on the temperature of the convective pulses, the amount of primary ¹²C mixed into the envelope by third dredge-up, and the extent of the partial mixing zone. Then we perform a detailed analysis of the reaction rates involved in the production of ¹⁹F and the effects of their uncertainties. We find that the major uncertainties are associated with the ¹⁴C(α, γ)¹⁸O and ¹⁹F(α, p)²²Ne reaction rates. For these two reactions we present new estimates of the rates and their uncertainties. In both cases the revised rates are lower than previous estimates. The effect of the inclusion of the partial mixing zone on the production of fluorine strongly depends on the very uncertain ¹⁴C(α, γ)¹⁸O reaction rate. The importance of the partial mixing zone is reduced when using our estimate for this rate. Overall, rate uncertainties result in uncertainties in the fluorine production of about 50% in stellar models with mass 3 M and of about a factor of 7 in stellar models of mass 5 M. This larger effect at high masses is due to the high uncertainties of the ¹⁹F(α, p)²²Ne reaction rate. Taking into account both the uncertainties related to the partial mixing zone and those related to nuclear reactions, the highest values of ¹⁹F enhancements observed in AGB stars are not matched by the models. This is a problem that will have to be revised by providing a better understanding of the formation and nucleosynthesis in the partial mixing zone, as well as in relation to reducing the uncertainties of the ¹⁴C(α, γ)¹⁸O reaction rate. At the same time, the possible effect of cool bottom processing at the base of the convective envelope should be included in the computation of AGB nucleosynthesis. This process could, in principle, help to match the highest ¹⁹F abundances observed by decreasing the C/O ratio at the surface of the star, while leaving the ¹⁹F abundance unchanged.