  In this paper, we are concerned with the computations of the $p$-rank of
curves in two different setups. We first work with complete intersection
varieties in $\mb{P}^n \text{ for}~n\ge 2$ and compute explicitly the action of
Frobenius on the top cohomology group. In case of curves and surfaces, this
information suffices to determine if the variety is ordinary. Next, we consider
curves on more general surfaces with $p_g(S) = 0 = q(S)$ such as Hirzebruch
surfaces and determine $p$-rank of curves on Hirzebruch surfaces.
