Huayi Chen has obtained his doctoral degree in mathematics from Ecole Polytechnique in 2006. He has been associate professor at Université Paris Diderot from 2008 to 2012, professor at Université Grenoble Alpes from 2012 to 2016, and then professor at Université Paris Cité from 2016 to 2023. From 2024, he works at Westlake University as chair professor in mathematics. His research interests include Arakelov geometry, geometry of numbers and convex geometry. Atsushi Moriwaki is a mathematician specializing in algebraic geometry and arithmetic geometry, especially Arakelov geometry. He earned his Ph.D. in Mathematics from Kyoto University in 1991 and went on to serve as a professor at Kyoto University from 2003 to 2023. In 2023, he was appointed as a professor emeritus at Kyoto University and started a new role as a professor at Chubu University.

• Introduction.- Review and Preliminaries.- Normed Graded Linear Series over a Trivially Valued Field.- Arithmetic Volumes over a General Adelic Curve.- Hilbert-Samuel Property.- Relative Ampleness and Nefness.- Global Adelic Space of an Arithmetic Variety.- Generically Big and Pseudo-effective Adelic Line Bundles.- Global Positivity Conditions.- Appendix A: Some Slope Estimates.