  We investigate the notion of the $p$-divisor for foliations on a smooth
algebraic surface defined over a field of positive characteristic $p$ and we
study some of their properties. We present a structure theorem for the
$p$-divisor of foliations in the projective plane and the Hirzebruch surfaces
where we show that, under certain conditions, such $p$-divisors are reduced.

Algebraic Geometry