The most objective of this booklet is to give the so-called birational Arakelov geometry, that are seen as an mathematics analog of the classical birational geometry, i.e., the research of huge linear sequence on algebraic types. After explaining classical effects in regards to the geometry of numbers, the writer starts off with Arakelov geometry for mathematics curves, and keeps with Arakelov geometry of mathematics surfaces and higher-dimensional types. The e-book comprises such basic effects as mathematics Hilbert-Samuel formulation, mathematics Nakai-Moishezon criterion, mathematics Bogomolov inequality, the lifestyles of small sections, the continuity of mathematics quantity functionality, the Lang-Bogomolov conjecture etc. furthermore, the writer provides, with complete information, the evidence of Faltings' Riemann-Roch theorem. necessities for interpreting this publication are the elemental result of algebraic geometry and the language of schemes.

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