  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. In general,
extant nucleosynthesis models overestimate the fluorine production by AGB stars
with respect to observations. In this paper we review the relevant nuclear
reaction rates involved in the fluorine production/destruction. We perform this
analysis on a model with initial mass M=2 M$_\odot$ and Z=0.001. We found that
the major uncertainties are due to the $^{13}$C($\alpha$,n)$^{16}$O, the
$^{19}$F($\alpha$,p)$^{22}$Ne and the $^{14}$N(p,$\gamma$)$^{15}$O reactions. A
change of the corresponding reaction rates within the present experimental
uncertainties implies surface $^{19}$F variations at the AGB tip lower than
10\%. For some $\alpha$ capture reactions, however, larger variations in the
rates of those processes cannot be excluded. Thus, we explore the effects of
the variation of some $\alpha$ capture rates well beyond the current published
uncertainties. The largest $^{19}$F variations are obtained by varying the
$^{15}$N($\alpha$,$\gamma$)$^{19}$F and the $^{19}$F($\alpha$,p)$^{22}$Ne
reactions. The analysis of some $\alpha$ capture processes assuming a wider
uncertainty range determines $^{19}$F abundances in better agreement with
recent spectroscopic fluorine measurements at low metallicity. In the framework
of the latter scenario the $^{15}$N($\alpha$,$\gamma$)$^{19}$F and the
$^{19}$F($\alpha$,p)$^{22}$Ne reactions show the largest effects on fluorine
nucleosynthesis. The presence of poorly known low energy resonances make such a
scenario, even if unlikely, possible. We plan to directly measure these
resonances.
