Incidence systems on Cartesian powers of algebraic curves

TL;DR

This paper proves that certain algebraic structures derived from algebraic curves can interpret a field, addressing a question posed by Zilber and advancing understanding of their model-theoretic properties.

Contribution

It demonstrates that non-locally modular reducts of the Zariski structure on algebraic curves interpret a field, revealing new connections between algebraic geometry and model theory.

Findings

Reducts of Zariski structures on algebraic curves interpret fields.

Addresses Zilber's question on the interpretability of fields.

Advances understanding of the model-theoretic complexity of algebraic curves.

Abstract

We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's.