Counting points of bounded height on del Pezzo surfaces

Abstract

del Pezzo surfaces are isomorphic to either P¹ x P¹ or P² blown up <i>a</i> times, where <i>a</i> ranges from 0 to 8. We will look at lines on del Pezzo surfaces isomorphic to P² blown up <i>a</i> times with <i>a</i> ranging from 0 to 6. We will show that when we count points of bounded height on one of these surfaces, the number of points on lines give us the primary growth order, but the secondary growth order calculates the number of points on the rest of the surface and hence is a better representation of the geometry of the surface.