Curve-excluding fields

If C is a curve over \mathbb{Q} with genus at least 2 and C(\mathbb{Q}) is empty, then the class of fields K of characteristic 0 such that C(K) = \varnothing has a model companion, which we call C\mathrm{XF} . The theory C\mathrm{XF} is not complete, but we characterize the completions. Using C\mathrm{XF} , we produce examples of fields with interesting combinations of properties. For example, we produce (1) a model-complete field with unbounded Galois group, (2) an infinite field with a decidable first-order theory that is not “large” in the sense of Pop, (3) a field that is algebraically bounded but not “very slim” in the sense of Junker and Koenigsmann, and (4) a pure field that is strictly NSOP _{4} , i.e., NSOP _{4} but not NSOP _{3} . Lastly, we give a new construction of fields that are virtually large but not large.