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The origin of carbon and the relative importance of massive and low- to intermediate-mass stars in producing it is still a matter of debate. We aim at better understanding the origin of carbon by studying the trends of [C\/H], [C\/Fe], and [C\/Mg] versus [Fe\/H] and [Mg\/H] for 2133 FGK dwarf stars from the fifth <jats:italic>Gaia<\/jats:italic>\u2013ESO Survey internal data release (GES iDR5). The availability of accurate parallaxes and proper motions from <jats:italic>Gaia<\/jats:italic> DR2 and radial velocities from GES iDR5 allows us to compute Galactic velocities, orbits, absolute magnitudes, and, for 1751 stars, Bayesian-derived ages. Three different selection methodologies have been adopted to discriminate between thin- and thick-disk stars. In all the cases, the two stellar groups show different [C\/H], [C\/Fe], and [C\/Mg] and span different age intervals, with the thick-disk stars being, on average, older than the thin-disk ones. The behaviors of [C\/H], [C\/Fe], and [C\/Mg] versus [Fe\/H], [Mg\/H], and age all suggest that C is primarily produced in massive stars. The increase of [C\/Mg] for young thin-disk stars indicates a contribution from low-mass stars or the increased C production from massive stars at high metallicities due to the enhanced mass loss. The analysis of the orbital parameters <jats:italic>R<\/jats:italic>\n                  <jats:sub>med<\/jats:sub> and <jats:inline-formula>\n                     <jats:tex-math>\n\n<\/jats:tex-math>\n                     <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"apjab5dc4ieqn1.gif\" xlink:type=\"simple\"\/>\n                  <\/jats:inline-formula> supports an \u201cinside\u2013out\u201d and \u201cupside\u2013down\u201d formation scenario for the disks of the Milky Way.<\/jats:p>","DOI":"10.3847\/1538-4357\/ab5dc4","type":"journal-article","created":{"date-parts":[[2020,1,8]],"date-time":"2020-01-08T14:13:11Z","timestamp":1578492791000},"page":"55","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":30,"title":["The Gaia\u2013ESO Survey: Carbon Abundance in the Galactic Thin and Thick 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