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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.

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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.

For details let me recommend you to take a look at "Integral Closure of Ideals, Rings and Modules" by Swanson and Huneke, chapters 5 and 14.

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