Local cohomology of ideals and the $R_n$ condition of Serre

Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $As{s}^{i}E=\{Q\in AssE\mid heightQ=i\}$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R\mathrm{/}P$ satisfies Serre’s condition $R_i$ then $As{s}^{g+i+1}{H}_P^{g+1}(R)$ is a finite set. As an application of our techniques we prove that if $P$ is a prime ideal in $R$ such that $(R\mathrm{/}P{)}_{\mathfrak{q}}$ is regular for any non-maximal prime ideal $\mathfrak{q}$ then $H_P^i(R)$ has finitely many associate primes for all $i$.