  We address the geometrical critical behavior of the two-dimensional Q-state
Potts model in terms of the spin clusters (i.e., connected domains where the
spin takes a constant value). These clusters are different from the usual
Fortuin-Kasteleyn clusters, and are separated by domain walls that can cross
and branch. We develop a transfer matrix technique enabling the formulation and
numerical study of spin clusters even when Q is not an integer. We further
identify geometrically the crossing events which give rise to conformal
correlation functions. This leads to an infinite series of fundamental critical
exponents h_{l_1-l_2,2 l_1}, valid for 0 </- Q </- 4, that describe the
insertion of l_1 thin and l_2 thick domain walls.
