# A comparison between obstructions to local-global principles over semiglobal fields

## TL;DR

This paper compares different types of local-global principles for rational points over semiglobal fields, showing how valuation-based principles imply those based on regular models, thus clarifying the relationships between various obstructions.

## Contribution

It establishes that local-global principles with respect to valuations imply those with respect to regular models in the context of semiglobal fields.

## Key findings

- Valuation-based local-global principles imply regular model-based principles.
- Comparison of obstructions arising from valuation theory and geometric models.
- Clarification of the hierarchy of local-global principles over semiglobal fields.

## Abstract

We consider local-global principles for rational points on varieties, in particular torsors, over one-variable function fields over complete discretely valued fields. There are several notions of such principles, arising either from the valuation theory of the function field, or from the geometry of a regular model of the function field. Our results compare the corresponding obstructions, proving in particular that a local-global principle with respect to valuations implies a local-global principle with respect to a sufficiently fine regular model.