In joint work with Daniel Hast, we recast the paper of Jon Keating and Zeev Rudnick "The variance of the number of prime polynomials in short intervals and in residue classes" by studying the geometry of these short intervals through an associated highly singular variety. We manage to recover their results for a a general class of arithmetic functions up to a constant and also obtain information about the higher moments. Recently work of Brad Rodgers in "Arithmetic functions in short intervals and the symmetric group" gives new insight into the geometry of our variety.