Let's say we have a finite subgroup $\Gamma \subseteq SL(2,\mathbb{C})$ and consider the quotient variety $\mathbb{C}^2/\Gamma$, which will have one of the well-known ADE or du Val surface singularities and can be embedded into $\mathbb{C}^3$ as a hypersurface with a singular point at the origin. These singularities have crepant resolutions where the exceptional fiber is a union of $\mathbb{P}^1$s connected according to an ADE Dynkin diagram. They can be obtained by repeatedly blowing up at singular points, starting with blowing up at the origin in $\mathbb{C}^3$.


When doing this it seems that after the first blowup, the exceptional curve or curves you get correspond to the natural representation of $\Gamma$ on $\mathbb{C}^2$ that is given by its inclusion into $SL(2,\mathbb{C})$. (This is under the McKay correspondence, which gives a one-to-one correspondence between nodes in the Dynkin diagram and nontrivial irreducible representations of $\Gamma$ over $\mathbb{C}$.) For instance, if you start with an $A\_n$ singularity, defined by $x^2+y^2+z^{n+1}=0$ in $\mathbb{C}^3$, and blow up at the origin, the exceptional divisor will be $X^2+Y^2=0$ in $\mathbb{P}^2$ which is two $\mathbb{P}^1$s meeting at a point. These two $\mathbb{P}^1$s correspond to the outermost nodes in the $A\_n$ Dynkin diagram which are dual representations of $\Gamma$, the cyclic group of order $n+1$ in this case, that you get from its natural action on $\mathbb{C}^2$. (The node or nodes for each $\Gamma$ that I'm describing are also the node(s) that you would connect to the additional node on an extended Dynkin diagram.) Although I haven't checked recently I believe this also works for all the other ADE groups. 


My question is, is there some kind of "natural" explanation for this? I was hoping this might be explained by one of the geometric interpretations of the McKay correspondence out there, like the Bridgeland-King-Reid "Mukai implies McKay" paper (which I don't thoroughly understand).



Resolving ADE singularities by blowing up
(Maybe this is an answer, maybe it merely moves the lump around under the carpet.) M. Artin showed ["On rational singularities of surfaces"] that for any rational surface singularity (in particular, for du Val singularities) the fundamental cycle $Z$ (by definition, the smallest non-zero effective divisor $Y$ supported on the exceptional locus of the minimal resolution $X$ such that $Y.C\le 0$ for all exceptional curves $C$ on $X$) is defined, as a scheme, by the maximal ideal of the original singularity. Therefore $X$ factors through the first blow-up and the exceptional curves that appear there are exactly those whose strict transforms on $X$ have strictly negative intersection with $Z$.


Now think of $Z$ merely as an element of the appropriate irreducible root lattice, bearing in mind that in this context the roots $\alpha$ have square (= self-intersection) $\alpha.\alpha$ equal to $-2$. Then $Z$ is characterized by the property that $Z$ is positive linear combination of the simple roots $\alpha\_i$, the intersection number $Z.\alpha\_i\le 0$ for each $i$ and $Z$ is minimal with respect to these conditions. Separate consideration of each of these (I don't know how to make this step "natural") shows that $Z$ equals the highest root ("plus grande racine"). Therefore the irreducible curve(s) that appear in the first blow-up correspond to the simple root(s) $\alpha\_j$ such that $Z.\alpha\_j<0$. In turn, further inspection of each Dynkin diagram shows that such $\alpha\_j$ correspond to the node(s) that "you would connect to the additional node on an extended Dynkin diagram", exactly as you say.


