  We present a class of reductions of M\"obius type for the lattice equations
known as Q1, Q2, and Q3 from the ABS list. The deautonomised form of one
particular reduction of Q3 is shown to exist on the $A_1^{(1)}$ surface which
belongs to the multiplicative type of rational surfaces in Sakai's
classification of Painlev\'e systems. Using the growth of degrees of iterates,
all other mappings that result from the class of reductions considered here are
shown to be linearisable. Any possible linearisations are calculated explicitly
by constructing a birational transformation defined by invariant curves in the
blown up space of initial values for each reduction.
