Valuative Trees in Arbitrary Characteristic and Applications to Log Canonical Thresholds

Abstract

Valuations naturally encode birational invariants of algebraic varieties. This thesis investigates the valuative tree—the valuative space centered at the affine plane over a field—via two complementary approaches. The first analyzes its tree structure through the observer’s topology and introduces two parameterizations, skewness and thinness, defined using the sequence of key polynomials (SKP). Although obtained from distinct constructions, thinness is shown to coincide with log-discrepancy functions through the universal dual graph. The second approach employs tools from non-Archimedean geometry. We review the Berkovich affine line over a complete valued field, with particular attention to Hahn series and completed Puiseux series; in these cases, the valuative tree embeds naturally into the Berkovich line, revealing characteristic-dependent phenomena. As an application, we use valuative trees to compute the log canonical thresholdof analytically irreducible plane curves. We conclude by discussing an alternative approach to defining log discrepancy functions in positive characteristic, along with the counterpart to the log canonical threshold—the F-pure threshold.