Another PhD student in the department (she works on quadratic forms) asked me something about surface singularities, and while explaining some of its interesting aspects (McKay correspondence, Hironaka's resolution of singularities, intersection graphs, ...) I came to realise that there is something important missing from my knowledge on rational surface singularities.

Sure, you can say that a (isolated) surface singularity $X$ is rational if its resolution $f\colon Y\to X$ has the property that $\mathrm{R}^1f_*(\mathcal{O}_Y)=0$, and hence the arithmetic genus of the surface doesn't change, and you can then prove that it doesn't matter which resolution you pick etc. But the interesting property of rational singularities is that you can determine whether they are rational purely in terms of the intersection graph. And this is where studying surface singularities really shines, as the example of McKay correspondence (and more general quotient singularities) shows!

Recall that from a resolution we can extract the following data, as by Zariski's main theorem you know that the inverse image of your singularity is connected of dimension 1. Hence it is a union of curves $E_i$. We then have numerical data:

• the genus of the irreducible components $E_i$
• the self-intersections $E_i^2$
• the intersection numbers $E_i\cdot E_j$

Taking as vertices $v_i$ the $E_i$ and $E_i\cdot E_j$ edges from $v_i$ to $v_j$, whilst assigning the genus and self-intersection as weights we obtain a weighted graph called the intersection graph, and allows to study surface singularities in a combinatorial way!

The first cool property about this intersection graph is that Mumford has proven that a collection of genera and intersection numbers is associated to a surface singularity if and only if the matrix obtained from the intersection numbers is negative-definite. How cool is that?

Recognising rational surface singularities

So let $f\colon Y\to X$ be a resolution of a rational surface singularity. You can moreover prove that for a rational surface singularity it is a bunch of $\mathbb{P}^1$'s intersecting each other in interesting ways. We can assume by contracting things that all the exceptional curves $E_i\cong\mathbb{P}^1$ have self-intersection $\leq -2$. Then there is the following theorem by Artin, from his On isolated rational singularities of surfaces.

Theorem (Artin) The surface singularity $X$ is rational if and only if for each divisor $Z=\sum_{i=1}^n\alpha_iE_i$ with $\alpha_i\geq 0$ supported on the exceptional locus the arithmetic genus is negative.

Sure, this is a nice characterisation, but somehow we'd like to decide whether a surface singularity is rational by only looking at a single divisor. To do so, Artin introduces the fundamental cycle, i.e. the minimal cycle supported on the exceptional locus such that $Z\geq\sum_{i=1}^nE_i$ and $Z\cdot E_i\leq 0$ for all $i=1,\dotsc,n$. To obtain the existence of such a cycle one needs that the intersection matrix $(E_i\cdot E_j)_{i,j}$ is negative definite, but we know that this is a result of Mumford as indicated above.

Theorem (Artin) The fundamental cycle for any surface singularity $X$ has the property that its arithmetic genus is positive. Moreover, a surface singularity is rational if and only if the arithmetic genus of the fundamental cycle is exactly zero.

So, starting from your intersection graph together with its multiplicities you can start looking for the fundamental cycle by computing $Z\cdot E_i$ and if it is strictly positive you add $E_i$ to your cycle, and once you have it you compute its arithmetic genus and conclude whether the singularity is rational or not!

For proofs of these things in a more up-to-date language, there is the article Combinatorics of rational singularities.

Some other facts

While reading about surface singularities the following things came up:

1. you can realise any (smooth projective irreducible) curve as the exceptional locus, by considering cones over curves
2. if you don't like the previous construction, there is a construction using Jacobians
3. the self-intersection of the fundamental cycle is the multiplicity, if this multiplicity is exactly -2 then we are in the case of rational double points or du Val singularities for which we have McKay correspondence, if this multiplicity is -3 Artin classified the rational triple points in his article on rational surface singularities, these are particular intersection graphs which are trees and contain exactly one rational curve whose self-intersection is -3.

This whole (rational) surface singularity stuff is exciting stuff, and it happens to have connections to noncommutative algebraic geometry as explained in section 5.4 of Joe Karmazyn's interesting preprint Quiver GIT for varieties with tilting bundles, as you can recover the resolution by a tilting bundle which you can construct exactly because the surface singularity is rational. This is a (vast) generalisation of the phenomena occurring for Kleinian singularities and McKay correspondence, and Joe gave a cool talk about all this at a workshop a few weeks ago. Now it's time to stop rambling about surface singularities.