Tim Browning, July 9, 2020

Title: Del Pezzo surfaces by degrees

Abstract: The arithmetic of del Pezzo surfaces gets harder as the degree decreases, with the main questions being about existence and distribution of rational points. Degree 1 del Pezzo surfaces can be embedded in weighted projective space and admit a natural elliptic fibration. On the one hand their arithmetic is very simple --- they always have a rational point --- but any significant piece of further information appears to lie beyond the veil... I shall survey what is known about them before discussing a new upper bound for the density of rational points of bounded height that uses a variant of the square sieve worked out by Lillian Pierce. This is joint work with Dante Bonolis.