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"For a connected smooth projective curve of genus g, global sections of any line bundle L with deg(L) 2g 1 give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry: We replace projective space by tropical projective space, and an embedding by a homeomorphism onto its image preserving integral structures (or equivalently, since is a curve, an isometry), which is called a faithful tropicalization. Let be an algebraically closed field which is complete with respect to a nontrivial nonarchimedean value. Suppose that is defined over and has genus g 2 and that is a skeleton (that is allowed to have ends) of the analytification an of in the sense of Berkovich. We show that if deg(L) 3g 1, then global sections of L give a faithful tropicalization of into tropical projective space. As an application, when Y is a suitable affine curve, we describe the analytification Y an as the limit of tropicalizations of an effectively bounded degree"--

Geometry, Algebraic. --- Tropical geometry. --- Algebraic geometry -- Tropical geometry -- Tropical geometry. --- Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Rigid analytic geometry. --- Algebraic geometry -- Cycles and subschemes -- Divisors, linear systems, invertible sheaves.

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The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell-Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole.

Mordell conjecture. --- Corbes algebraiques