On the coregularity of del Pezzo surfaces with du Val singularities

We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, 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 some special properties. We also prove results about 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 $0$, while del Pezzo surfaces with positive coregularity enjoy some special properties.