# General conditions for normality of blow-up

Let $$X$$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $$Z$$ in $$X$$ such that the reduced scheme associated to the blow-up of $$X$$ along $$Z$$ is once again normal. Any reference will be most welcome.

Let $$X=Spec(R)$$. Blowing-up $$Z=V(I)$$ is the same as to look at $$Proj$$ of the graded ring $$R[It]=\oplus_{j\geqslant 0} I^jt^j\subset R[t]$$, the Rees ring associated to $$I$$.
Assume $$R$$ is a domain, integrally closed inside its fraction field $$K$$. In this case the integral closure of $$R[It]$$ inside its fraction field is
$$\overline{R[It]}=\oplus_{j\geqslant 0} \overline{I^j}t^j$$. Here, if $$J\subset R$$ is an ideal of $$R$$ the notation $$\overline{J}$$ is the integral closure of the ideal $$J$$. This is again an ideal inside $$R$$.
If $$R$$ is of dimension $$2$$ and $$I=\overline{I}$$ then results due to Zariski (if $$R$$ is a rlr) show that $$I^j=\overline{I^j}$$ for all $$j$$ and therefore $$R[It]$$ is integrally closed and you have the condition that ensures normality. These results were later generalized by Michael Artin to normal local rings of dimension $$2$$.
I should also say that there are very explicit combinatorial criteria (in terms of Newton polygons) to determine whether a monomial ideal $$I\subset\mathbb{C}[[x,y]]$$ or $$\mathcal{O}_{\mathbb{C}^2,0}$$ is integrally closed.