Valuation theoretic methods in the birational geometry of algebraic varieties

TL;DR

This paper develops valuation formulas for rational differential forms and log discrepancies in algebraic geometry, extending classical concepts to arbitrary Abhyankar places, thereby broadening the adjunction machinery in birational geometry.

Contribution

It introduces a valuation formula for differential forms at Abhyankar places and generalizes log discrepancies and adjunction to these places, expanding the tools in birational geometry.

Findings

Valuation formula for rational top differential forms at Abhyankar places

Definition of log discrepancies for arbitrary Abhyankar places

Generalization of the adjunction machinery to these places

Abstract

In this paper, we give a valuation formula for rational top differential forms of function fields in characteristic zero for arbitrary Abhyankar places generalizing the classical valuation at prime divisors. This enables us to define log discrepancies for log pairs for arbitrary Abhyankar places. If the Abhyankar place has dimension greater than zero we restrict rational top differential forms with valuation zero to the residue field of the Abhyankar place, generalizing the classical restriction of a top differential form with a simple pole along a smooth divisor. This opens up the door to generalize the classical adjunction machinery to arbitrary Abhyankar places.