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.

This monograph presents new research on Arakelov geometry over adelic curves, a novel theory of arithmetic geometry developed by the authors. It explores positivity conditions and establishes the Hilbert-Samuel formula and the equidistribution theorem in the context of adelic curves. Connections with several classical topics in Arakelov geometry and Diophantine geometry are highlighted, such as the arithmetic Hilbert-Samuel formula, positivity of line bundles, equidistribution of small subvarieties, and theorems resembling the Bogomolov conjecture. Detailed proofs and explanations are provided to ensure the text is accessible to both graduate students and experienced researchers.