Generalized Eckardt points on del Pezzo surfaces of degree 1

Abstract

We study intersections of exceptional curves on del Pezzo surfaces of degree 1, motivated by questions in arithmetic geometry. Outside characteristics 2 and 3, at most 10 exceptional curves can intersect in a point. We classify the different ways in which 10 exceptional curves can intersect, construct a new family of surfaces with 10 exceptional curves intersecting in a point, and discuss strategies for finding more such examples.