  It is well appreciated that the description of overadiabatic convection
affects the structure of the envelopes of luminous asymptotic giant branch
(AGB) stars in the phase of ``hot bottom burning '' (HBB). We stress that this
important uncertainty in the modeling plays a role which is much more dramatic
than the role which can be ascribed, e.g., to the uncertainty in the nuclear
cross-sections. Due to the role tentatively attributed today to the HBB
nucleosynthesis as the site of self-enrichment of Globular Clusters stars, it
is necessary to explore the difference in nucleosynthesis obtained by different
prescriptions for convection. We present results of detailed evolutionary
calculations of the evolution of stars of intermediate mass during the AGB
phase. We follow carefully the nucleosynthesis at the base of the external
convective region, showing that very different results can be obtained
according to the presciption adopted to find out the temperature gradient
within the instability regions. We discuss the uncertainties in the yields of
the various chemical species and the role which these sources can play as
polluters of the interstellar medium.
