On the coregularity of del Pezzo surfaces with du Val singularities

We compute the coregularity of del Pezzo surfaces with du Val singularities. To do this we study the relation between del Pezzo surfaces of degree 1 and elliptic fibrations. It turns out that del Pezzo surfaces with positive coregularity correspond to isotrivial elliptic fibrations with certain special properties. We also prove results about the coregularity of del Pezzo surfaces over non-algebraically closed fields of characteristic 0. Our results confirm the expectation that ‘most’ del Pezzo surfaces have coregularity 0. Bibliography: 32 titles.