First Coefficient ideals and $R_1$ property of Rees algebras

Abstract

Let $(A,\mathfrak{m})$ be an excellent normal local ring of dimension $d \geq 2$ with infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal. Then the following assertions are equivalent: (i) The extended Rees algebra $A[It, t^{-1}]$ is $R_1$. (ii) The Rees algebra $A[It]$ is $R_1$. (iii) $Proj(A[It])$ is $R_1$. (iv) $(I^n)^* = (I^n)_1$ for all $n \geq 1$. Here $(I^n)^*$ is the integral closure of $I^n$ and $(I^n)_1$ is the first coefficient ideal of $I^n$.