Rees algebras and $p_g$-ideals in a two-dimensional normal local domain

The authors introduced the notion of $p_g$-ideals for two-dimensional excellent normal local domain over an algebraicaly closed field in terms of resolution of singularities. In this note, we give several ring-theoretic characterization of $p_g$-ideals. For instance, an $m$-primary ideal $I\subset A$ is a $p_g$-ideal if and only if the Rees alegbra $\mathcal{R}(I)$ is a Cohen-Macaulay normal domain.