Context. The origin of fluorine is a longstanding problem in nuclear astrophysics. It is widely recognized that asymptotic giant branch (AGB) stars are among the most important contributors to the Galactic fluorine production. Aims. In general, extant nucleosynthesis models overestimate the fluorine production by AGB stars with respect to observations. Although those differences are rather small at solar metallicity, low metallicity AGB stellar models predict fluorine surface abundances up to one order of magnitude larger than the observed ones. Methods. As part of a project devoted to reducing the uncertainties in the nuclear physics that affect the nucleosynthesis in AGB stellar models, we review the relevant nuclear reaction rates involved in the fluorine production or destruction. We perform this analysis on a model with initial mass M = 2 M and Z = 0.001. Results. We found that the major uncertainties are due to the 13 C(α,n) 16 O, the 19 F(α,p) 22 Ne, and the 14 N(p,γ) 15 O reactions. A change in the corresponding reaction rates within the present experimental uncertainties implies surface 19 F variations at the AGB tip lower than 10%, thus much smaller than observational uncertainties. For some α capture reactions, however, cross sections at astrophysically relevant energies are determined on the basis of nuclear models, in which some low-energy resonance parameters are very poorly known. Thus, larger variations in the rates of those processes cannot be excluded. That being so, we explore the effects of the variation in some α capture rates well beyond the current published uncertainties. The largest 19 F variations are obtained by varying the 15 N(α,γ) 19 F and the 19 F(α,p) 22 Ne reactions. Conclusions. The currently estimated uncertainties of the nuclear reaction rates involved in the production and destruction of fluorine produce minor 19 F variations in the ejecta of AGB stars. Analysis of some α capture processes that assume a wider uncertainty range determines 19 F abundances in better agreement with recent spectroscopic fluorine measurements at low metallicity. In the framework of this scenario, the 15 N(α,γ) 19 F and the 19 F(α,p) 22 Ne reactions show the strongest effects on fluorine nucleosynthesis. The presence of poorly known low-energy resonances make such a scenario possible, even if it is unlikely. We plan to measure these resonances directly.