We investigate the conjecture that the horizon of an index alpha fractional Brownian surface has (almost surely) the same Hölder exponents as the surface itself, with corresponding relationships for fractal dimensions. We establish this formally for the usual Brownian surface (where alpha = 1/ 2 ), and also for other alpha, 0 < alpha< 1, assuming a hypothesis concerning maxima of index alpha Brownian motion. We provide computational evidence that the conjecture is indeed true for all alpha.