Polarized Halphen Surfaces and du Val Curves

A genus-g du Val curve is a degree-3g plane curve having 8 points of multiplicity g, one point of multiplicity g-1, and no other singularity. In a joint work with A. Bruno, G. Farkas and G. Saccà, we prove that a general du Val curve is Brill-Noether-Petri curve. This also gives examples of BNP curves defined over Q. In a joint work (in progress) with A. Bruno, we prove that the corank of the Gauss-Wahl map of a du Val curve is equal to one. The two results together show that the characterization of curves with non-surjective Gauss-Wahl map as hyperplane section of K3 surfaces and limits thereof (obtained jointly with A. Bruno and E. Sernesi) is optimal. This is done via the study of polarized Halphen surfaces.