  The Mott insulating phase of the parent compounds is frequently taken as
starting point for the underdoped high-$T_c$ cuprate superconductors. In
particular, the pseudogap state is often considered as deriving from the Mott
insulator. In this work, we systematically investigate different weakly-doped
Mott insulators on the square and triangular lattice to clarify the
relationship between the pseudogap and Mottness. We show that doping a
two-dimensional Mott insulator does not necessarily lead to a pseudogap phase.
Despite its inherent strong-coupling nature, we find that the existence or
absence of a pseudogap depends sensitively on non-interacting band parameters
and identify the crucial role played by the van Hove singularities of the
system. Motivated by a SU(2) gauge theory for the pseudogap state, we propose
and verify numerically a simple equation that governs the evolution of
characteristic features in the electronic scattering rate.
