A note on the definability of genus for Zariski geometries

TL;DR

This paper introduces a genus concept for Zariski geometries, generalizes classical theorems, and demonstrates that this genus cannot be defined by a first-order formula within the full language of such geometries.

Contribution

It proposes a new notion of genus for Zariski geometries and extends classical theorems to this setting, revealing limitations of definability.

Findings

Generalized Riemann--Hurwitz and Hurwitz theorems for Zariski geometries

The genus notion is not first-order definable in the full language

Provides a bridge between classical algebraic geometry and model theory

Abstract

In this work we propose a notion of genus in the context of Zariski geometries and we obtain natural generalizations of the Riemann--Hurwitz Theorem and the Hurwitz Theorem in the context of very ample Zariski geometries. As a corollary, we show that such notion of genus cannot be first-order definable in the full language of a Zariski geometry.