  To study the progenitors of PNe at the tip of the Planetary Nebula Luminosity
Function (PNLF), we obtained the deepest existing spectra of a sample of PNe in
the galaxy M31. Precise chemical abundances allow us to confront the
theoretical yields for Asymptotic Giant Branch (AGB) stellar masses and
metallicities expected at the bright end of the PNLF. Central star masses of
the sampled PNe provide direct information on the controversial origin of the
universal cutoff of the PNLF.
  Using the 10.4m Gran Telescopio Canarias (GTC) optical spectra of nine bright
M31 PNe were obtained: four of them at the tip of the PNLF, and the other five
some 0.5 magnitudes fainter. A control sample of 21 PNe with previous GTC
spectra from literature is also included. Their physical properties and
chemical abundances (He, N, O, Ar, Ne and S) are analysed. The central star
masses are estimated with Cloudy modelling using the most recent evolutionary
tracks.
  The studied PNe show a remarkable uniformity in all their nebular properties,
the brightest PNe showing relatively large electron densities. Stellar
characteristics also span a narrow range: ${<L_{*}/L_{\odot}> = 4300 \pm 310,
<Teff> = 122000 \pm 10600 K}$ for the central stars of the four brightest PNe,
and ${<L_{*}/L_{\odot}> = 3300 \pm 370, < Teff> = 135000 \pm 26000 K}$ for
those in the control set. This groups all the brightest PNe at the location of
maximum temperature in the post-AGB tracks for stars with initial masses ${M =
1.5 M_{\odot}}$.
  These figures provide robust observational constraints about the stellar
progenitors producing the PNLF cutoff in a star-forming galaxy such as M31,
where a large range of initial masses is in principle available. Inconsistency
is found, however, in the computed N/O abundance ratios of five nebulae, which
are 1.5 to 3 times larger than predicted by the existing models for stars of
these masses.
