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I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":

Here the relevant excerpt:

enter image description here Remark: this is the continuation of the proof of Thm 4 (page 132) where we assumed that $\pi: V \to \bar{V}= Spec(A)$ is a birational map, $A$ henselian and $X \subset V$ reduced curve on normal surface $V$ with components $X_i$.

The question is if $D$ is apositive divisor which meets $X_i$ transversally why the intersections of $D$ with $X$ correspond one to one with connected components of the support of $D$? What property of Henselian rings is exploited here? I'm just familar with the lifting story of polynomials in $A[X]$ which have non multiple roots in $(A/m)[X]$.

But I don't see how it can be applied here. Is there another characterization for Henselians with more "geometric" flavour?

Remark: Some weeks ago I asked another question about properties of Henselian local rings: Henselian Schemes

Does the mysterious property of Henselian which I'm looking for in this thread provide an argument for my former thread?

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