Bertini's theorem states that the general curve of a system which is linearly dependent on a certain number of given irreducible algebraic curves will not have a singular point which is not fixed for all the curves of the system.
Over a finite field, a nonempty open set need not contain a rational point. Poonen (2004) proved a finite-field version in which the polynomial degree of the intersecting hypersurface grows while the embedding of a smooth variety is fixed. The related Baker's question asks whether varying the embedding eventually guarantees a smooth hyperplane section.
