Hyperbolicity and model-complete fields
Michał Szachniewicz, Jinhe Ye
TL;DR
This work generalizes the curve case of model-companion existence for theories of fields avoiding a variety to higher dimensions by introducing excludability. Using Hilbert schemes and intersection theory, it proves a main existence theorem for $V ext{XF}$ under boundedness and indecomposability assumptions, and shows that the resulting theory enjoys strong model-theoretic and field-theoretic properties (QE with Sol predicates, algebraic and model-theoretic closures coinciding, Hilbertian PAC finite extensions, and an $ extomega$-free Galois group) while remaining NSOP$_4$ and TP$_2$. The authors further develop a framework for indecomposable $V$, derive explicit consequences for the absolute closures of models, and discuss decidability in the incomplete theory under computable conditions. They also provide a characteristic-$p$ counterexample illustrating limits of excludability and conclude with open questions about rigidity, broader applicability, and potential group-action extensions. Overall, the paper advances the program linking hyperbolicity notions to model-theoretic tameness by elevating results from curves to higher-dimensional excludable varieties.
Abstract
We study model-complete fields that avoid a given quasi-project variety $V$. There is a close connection between hyperbolicity of $V$ and the existence of the model companion for the theory of characteristic-zero fields avoiding rational points on $V$. This gives a model theoretic notion of hyperbolicity that we call excludability. In particular, we show that if $V$ is a Brody hyperbolic projective variety over $\mathbb{Q}$ with $V(\mathbb{Q}) = \varnothing$, then the model companion, called $V\XF$, exists. We also study some model-theoretic properties of $V\mathrm{XF}$. This extends the results for curves by Will Johnson and the second author.