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S stars are late-type giants that are transition objects between M-type starsand carbon stars on the asymptotic giant branch (AGB). They are classified intotwo types: intrinsic or extrinsic, based on the presence or absence oftechnetium (Tc). The Tc-rich or intrinsic S stars are thermally-pulsing(TP-)AGB stars internally producing s-process elements (including Tc) which arebrought to their surface via the third dredge-up (TDU). Tc-poor or extrinsic Sstars gained their s-process overabundances via accretion of s-process-richmaterial from an AGB companion which has since turned into a dim white dwarf.Our goal is to investigate the evolutionary status of Tc-rich S stars bylocating them in a Hertzsprung-Russell (HR) diagram using the results of Gaiaearly Data Release 3 (EDR3). We combine the current sample of 13 Tc-rich starswith our previous studies of 10 Tc-rich stars to determine the observationalonset of the TDU in the metallicity range [-0.7; 0]. We also compare ourabundance determinations with dedicated AGB nucleosynthesis predictions. Thestellar parameters are derived using an iterative tool which combines HERMEShigh-resolution spectra, accurate Gaia EDR3 parallaxes, stellar evolutionmodels and tailored MARCS model atmospheres for S-type stars. Using thesestellar parameters we determine the heavy-element abundances by line synthesis.In the HR diagram, the intrinsic S stars are located at higher luminositiesthan the predicted onset of the TDU. These findings are consistent with Tc-richS stars being genuinely TP-AGB stars. The comparison of the derived s-processabundance profiles of our intrinsic S stars with the nucleosynthesispredictions provide an overall good agreement. Stars with highest [s/Fe] tendto have the highest C/O ratios.

S stars and s-process in the Gaia era II. Constraining the luminosity of the third dredge-up with Tc-rich S stars