Orthogonal projections in the plane over prime order fields

Abstract: We give an upper bound on the number of exceptional orthogonal projections of a small set of points in a plane over a prime order field, which makes progress toward a conjecture of Chen (2018). This theorem relies on a new upper bound on the number of incidences between an arbitrary set of lines and a set of points contained in a reasonably small pencil of lines. We also prove a new upper bound on the number of incidences between points and lines when both are Cartesian products in an appropriate sense.