Factorial residues modulo a prime: beyond the square-root bound

For a prime $p$, let $A_p=\{k!\pmod p:1\leq k<p\}$. We prove $|A_p|\gg p^{8/15}$, improving the general lower bound $(\sqrt{2}-o(1))p^{1/2}$. The proof begins with the identity $(n+2)!=(n+1)!+((n+1)!)^2/n!$ in $\mathbb{F}_p$, which produces many incidences for a family of fractional-linear maps. After Cauchy--Schwarz, the transition maps between two members of this family become affine lines, with multiplicity at most two. The Cartesian-product point-line incidence theorem of Stevens and de Zeeuw then yields the exponent $8/15$.