We investigate the discrepancy between oxygen abundance estimations fornarrow-line regions (NLRs) of Active Galactic Nuclei (AGNs) type Seyfert 2derived by using direct estimations of the electron temperature (Te-method) andthose derived by using photoionization models. In view of this, observationalemission-line ratios in the optical range (3000 < \lambda(\AA) < 7000) ofSeyfert 2 nuclei compiled from the literature were reproduced by detailedphotoionization models built with the Cloudy code. We find that the deriveddiscrepancies are mainly due to the inappropriate use of the relations betweentemperatures of the low (t2) and high (t3) ionization gas zones derived for HII regions in AGN chemical abundance studies. Using a photoionization modelgrid, we derived a new expression for t2 as a function of t3 valid for Seyfert2 nuclei. The use of this new expression in the AGN estimation of the O/Habundances based on Te-method produces O/H abundances slightly lower (about 0.2dex) than those derived from detailed photoionization models. We also find thatthe new formalism for the Te-method reduces by about 0.4 dex the O/Hdiscrepancies between the abundances obtained from strong emission-linecalibrations and those derived from direct estimations.

The stellar-to-halo mass relation (SHMR) is one of the main sources ofinformation we have on the connection between galaxies and their dark matterhaloes. Here we analyse in detail two popular forms of the SHMR, M*/Mh vs Mhand M*/Mh vs M*, and compare them with another physically motivated scalingrelation, M*/Mh vs GMh/j*sigmahat*. Although this relation cannot predict thehalo mass explicitely, it connects the stellar mass fraction to fundamentalgalaxy properties such as specific angular momentum (j*) and velocitydispersion (sigmahat*) via disc gravitational instability. Our detailedcomparative analysis is based on one of the largest sample of galaxies withboth high-quality rotation curves and near-infrared surface photometry, andleads to the following results: (i) M*/Mh vs Mh and M*/Mh vs M* are not justtwo alternative parametrizations of the same relation, but two significantlydifferent relations; (ii) M*/Mh vs GMh/j*sigmahat* outperforms the two popularrelations in terms of tightness, correlation strength and significance; (iii)j* and sigmahat* play an equally important role in our scaling relation, and itis their interplay that constrains M*/Mh so tightly; (iv) the evolution ofM*/Mh, j* and sigmahat* is regulated by disc gravitational instability: whenM*/Mh varies, j* and sigmahat* also vary as predicted by our scaling relation,thus erasing the memory of such evolution. This implies that the process ofdisc gravitational instability is intriguingly uniform across disc galaxies ofall morphological types: from lenticulars to blue compact dwarfs. Inparticular, the cosmic variance of Toomre's Q is 0.2 dex, a universal value forboth stars and atomic gas.