Group Chunks in Model Theory and Algebraic Geometry

Abstract

We formulate a group chunk theorem in the context of sheaves on sites which generalizes many similar results in model theory and algebraic geometry. Secondly, we develop an algebro-geometric analogue of Hrushovski's method of producing a group chunk from germs of definable functions on stationary types. The use of the model-theoretic tools of canonical bases and elimination of imaginaries is replaced with the use of Hilbert schemes to study ``canonical'' families of rational morphisms, allowing us to extend the previously known results over more general base schemes. The proofs of these results involve some technical work which may be of independent interest. First, we study partial morphisms in arbitrary categories, and show that a presheaf of partial magmas on a small category admits a universal morphism to a group. On the model theory side, we show that type-definable sets of $M^{eq}$ can be interpreted as sheaf quotients of type-definable sets. On the algebraic geometry side, we develop a theory of rational morphisms and families of rational morphisms of schemes over an arbitrary base.