Let $(R, \mathfrak{m})$ be a Noetherian normal local domain and $I$ an $R$-ideal. Write $X$ for the normalization of $\mathrm{Proj}(R[It])$ and $E$ for the effective Cartier divisor defined by the invertible ideal sheaf $I\mathcal{O}_X$. We may place suitable hypothesis on $R$ (excellent / essentially of finite type over a field / that completion has no nilpotents) to guarantee that $X$ is of finite type over $\mathrm{Spec}(R)$.
One sees from the exact sequence $0 \to I\mathcal{O}_X \to \mathcal{O}_X \to \mathcal{O}_E \to 0$ that $\mathrm{H}^0(E, \mathcal{O}_E) \supseteq R/\overline{I}$, where $\overline{I}$ denotes the integral closure of $I$. Is it true that $\mathrm{H}^0(E, \mathcal{O}_E) = R/\overline{I}$? I am specifically interested in the case: $\dim R = 2$, $I$ is $\mathfrak{m}$-primary and $R$ is not pseudorational. (If $R$ is pseudorational, then the answer is positive.)
Any help/reference will be appreciated.
Thanks; Manoj.
