Galactic planetary nebulae with precise nebular abundances as a tool to
  understand the evolution of asymptotic giant branch stars
  We present nucleosynthesis predictions (HeCNOCl) from asymptotic giant branch
(AGB) models, with diffusive overshooting from all the convective borders, in
the metallicity range Z/4 < Z < 2Zsun. They are compared to recent precise
nebular abundances in a sample of Galactic planetary nebulae (PNe) that is
divided among double-dust chemistry (DC) and oxygen-dust chemistry (OC)
according to the infrared dust features. Unlike the similar subsample of
Galactic carbon-dust chemistry PNe recently analysed by us, here the individual
abundance errors, the higher metallicity spread, and the uncertain dust
types/subtypes in some PNe do not allow a clear determination of the AGB
progenitor masses (and formation epochs) for both PNe samples; the comparison
is thus more focussed on a object-by-object basis. The lowest metallicity OC
PNe evolve from low-mass (~1 Msun) O-rich AGBs, while the higher metallicity
ones (all with uncertain dust classifications) display a chemical pattern
similar to the DC PNe. In agreement with recent literature, the DC PNe mostly
descend from high-mass (M > 3.5 Msun) solar/supersolar metallicity AGBs that
experience hot bottom burning (HBB), but other formation channels in low-mass
AGBs like extra mixing, stellar rotation, binary interaction, or He
pre-enrichment cannot be disregarded until more accurate C/O ratios would be
obtained. Two objects among the DC PNe show the imprint of advanced CNO
processing and deep second dredge-up, suggesting progenitors masses close to
the limit to evolve as core collapse supernovae (above 6 Msun). Their actual
C/O ratio, if confirmed, indicate contamination from the third dredge-up,
rejecting the hypothesis that the chemical composition of such high-metallicity
massive AGBs is modified exclusively by HBB.
