Invariants of algebraic varieties over imperfect fields

TL;DR

This paper introduces four invariants for algebraic varieties over imperfect fields, explores their properties, and applies them to curves, including genus change and boundedness results, with explicit computations.

Contribution

It defines new invariants for varieties over imperfect fields, analyzes their properties, and applies them to specific cases like curves, providing new insights and explicit examples.

Findings

Established fundamental properties of the invariants.

Proved a genus change formula for curves.

Showed boundedness of non-smooth genus one curves.

Abstract

We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.