Turing degrees of isomorphism types of geometric objects

TL;DR

This paper explores the computability-theoretic properties of geometric objects like ringed spaces and schemes, demonstrating the diversity of Turing degrees associated with their isomorphism types.

Contribution

It introduces the study of Turing degrees in the context of geometric structures, showing that all degrees can occur and some structures lack a least degree.

Findings

Any Turing degree can be the least degree of an isomorphic copy.

Some structures do not have a least degree.

The study bridges computability theory and algebraic geometry.

Abstract

We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures may fail to have a least degree.