Electron temperature relations and the direct N, O, Ne, S and Ar abundances of 49959 star-forming galaxies in DESI Data Release 2

We present the largest direct-method abundance catalogue of galaxies to date, containing measurements of 49$$959 star-forming galaxies at $z<0.96$ from DESI Data Release 2. By directly measuring electron temperatures across multiple ionization zones, we provide constraints on a number of electron temperature relations finding good consistency with previous literature relations. Using these temperature measurements, we derive reliable abundances for N, O, Ne, S and Ar and measure the evolution of abundances and abundance ratios of as a function of metallicity and other galaxy properties. Our measurements include direct oxygen abundances for 49$$766 galaxies, leading to the discovery of the two most metal-poor galaxies in the nearby Universe, with oxygen abundances of $12+\log (\mathrm{O}\mathrm{/}\mathrm{H})=6.77_{-0.03}^{+0.03}\mathrm{d}\mathrm{e}\mathrm{x}$ (1.2% ${\mathrm{Z}}_{\odot }$) and $12+\log (\mathrm{O}\mathrm{/}\mathrm{H})=6.81_{-0.04}^{+0.04}\mathrm{d}\mathrm{e}\mathrm{x}$ (1.3% ${\mathrm{Z}}_{\odot }$). We identify a rare outlier population of 139 galaxies with high N/O ratios at low metallicity, reminiscent of galaxy abundances observed in the early Universe. We find these high N/O galaxies are more massive than typical galaxies at the same metallicity. We find the Ne/O ratio is constant at low metallicity but increases significantly at $12+\mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{O}\mathrm{/}\mathrm{H})>8.105\pm 0.004$ dex. We show that the S/O and Ar/O abundance ratios are strongly correlated, consistent with the expected additional Type Ia enrichment channel for S and Ar. In this work we present an initial survey of the key properties of the sample, with this dataset serving as a foundation for extensive future work on galaxy abundances at low redshift.