On $p_g$-ideals in positive characteristic

Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two containing a field $k\cong A\mathrm{/}\mathfrak{m}$. 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.