Erratic birational behavior of mappings in positive characteristic

TL;DR

This paper investigates the complex and unpredictable behavior of birational mappings in positive characteristic, showing that stable forms can vary erratically after blowing up, unlike the predictable forms in characteristic zero.

Contribution

It demonstrates that in positive characteristic, stable forms of birational mappings can behave erratically, providing explicit examples in Artin-Schreier extensions.

Findings

Stable forms exist after blowing up in positive characteristic.

Forms can vary dramatically even within stable configurations.

Examples constructed in towers of defect Artin-Schreier extensions.

Abstract

Birational properites of generically finite morphisms $X\rightarrow Y$ of algebraic varieties can be understood locally by a valuation of the function field of $X$. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in towers of defect Artin-Schreier extensions which can have any prescribed distance.