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 without rational points on V, extending the results for curves by Will Johnson and Jinhe Ye. In particular, we show that if V is a Brody hyperbolic projective variety over Q with V(Q) =, then the model companion, called VXF, exists. We also study some model-theoretic properties of VXF.
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Szachniewicz et al. (2024) studied this question.
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