  Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two
containing a field $k \cong A/\mathfrak{m}$. An $\mathfrak{m}$-primary ideal
$I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a Cohen-Macaulay normal
domain. If $k$ is algebraically closed then Okuma, Watanabe and Yoshida proved
that $A$ has $p_g$-ideals and furthermore product of two $p_g$-ideals is a
$p_g$ ideal. In a previous paper we showed that if $k$ has characteristic zero
then $A$ has $p_g$-ideals. In this paper we prove that if $k$ is perfect field
of positive characteristic then also $A$ has $p_g$ ideals.
