Graded linear series of line bundles are natural objects in algebraic geometry. Classically, cohomological methods are standard in the study of the total graded linear series of a line bundle with suitable positivity conditions such as ampleness or numerical effectivity. In general case where the graded linear series is partial or the corresponding line bundle is not sufficiently positive, the application of cohomological methods is ofter quite subtle. In this talk, I will explain a new approach to the study of graded linear series, which is based on function field arithmetic.

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