In a joint work with A. Bruno, G. Farkas and G. SaccA we use surfaces having one elliptic singularity, and canonical curves as hyperplane sections, to construct explicit examples of smooth Brill-Noether-Petri curves, of any given genus, defined over the rational numbers. This provides a negative answer to a question raised by Harris and Morrison. The work stems from a joint result with A. Bruno and E. Sernesi where we give a complete characterization of Brill-Noether-Petri curves that are hyperplane sections of K3 surfaces or of a limit thereof.