# Question about Local Henselian Rings

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:

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?