Overview
The study focuses on the local stability of Frobenius and Artin–Schreier base-change within the context of a family of pairs $(X, \Delta)$ defined over a smooth curve $C$. The foundational characteristic of the mathematical setting is a positive integer $p \neq 0$. The primary objective involves a comparative analysis of these two types of base-change, specifically examining the discrepancies of divisors.
Research Context
The investigation is situated within the broader field concerning the behavior of mathematical structures under specific transformations, known as base-changes. The base-changes under consideration are Artin–Schreier and Frobenius covers of the curve $C$. The theoretical framework for this analysis incorporates ramification data, specifically for Artin–Schreier covers, and degree $p$ purely inseparable extensions of discrete valuation rings (DVRs). This context suggests an exploration of how geometric or algebraic properties, represented by divisors, change or are preserved across these transformations.
Approach
The research approach involves studying the discrepancies of divisors as they relate to base-changes $(X, \Delta) \to C$. This study is conducted along two distinct types of covers: Artin–Schreier covers and Frobenius covers of $C$. The analytical tools employed for this investigation include ramification data, which is utilized for the Artin–Schreier case. Additionally, the methodology incorporates the examination of degree $p$ purely inseparable extensions of DVRs. These tools are applied to understand the behavior and properties of the discrepancies of divisors under these specific base-change operations.
Findings
The core finding of this research is the recovery of a specific result previously established by Hu and Zong. This result pertains to the permanence of local stability for the types of base-change under investigation, namely Frobenius and Artin–Schreier base-change. The recovery of this result indicates that the methodology employed in the current study, involving the analysis of divisor discrepancies through ramification data and degree $p$ purely inseparable extensions of DVRs, substantiates or re-establishes the earlier finding regarding the enduring nature of local stability under these conditions.
Why This Matters
This research contributes to the understanding of local stability properties within algebraic geometry, particularly concerning families of pairs over smooth curves in positive characteristic. By recovering a result from Hu and Zong, the study validates existing theoretical frameworks and analytical methods for understanding the behavior of divisors under specific types of base-changes. This reinforces the foundational knowledge base in an area of mathematics that deals with the intrinsic properties of algebraic varieties and their transformations.