In this paper we study hypergraphs definable in an algebraically closed field. Our goal is to show, in the spirit of the so-called transference principles in extremal combinatorics, that if a given algebraic hypergraph is "dense" in a certain sense, then a generic low-dimensional subset of its vertices induces a subhypergraph that is also "dense." (For technical reasons, we only consider low-dimensional subsets that are parameterized by rational functions.) Our proof approach is inspired by the hypergraph containers method, developed by Balogh, Morris, and Samotij and independently by Saxton and Thomason (although adapting this method to the algebraic setting presents some unique challenges that do not occur when working with finite hypergraphs). Along the way, we establish a natural generalization of the classical dimension of fibers theorem in algebraic geometry, which is interesting in its own right.