Posted by: lewallen | November 18, 2008

## Surface Singularities and Dynkin Diagrams

In this post, I want to talk about a really cool and surprising connection between simple Lie algebras (via Dynkin diagrams) and certain singularities of quotient varieties. More generally, this defines a correspondence between simple Lie algebras and some discrete subgroups of $GL(2,\mathbb{C})$, which is a small part of Arnol’d’s “mathematical trinities.” At the end of the post, I include Arnol’d’s amazing table, with a link to his exposition of it. Thanks to Boris Alexeev and Tamar Friedman for introducing this material to me. For the most part, I follow online notes by Miles Reid. The singularities themselves (rather than Arnol’d’s table) are standard fair in algebraic surface theory, often under the heading of “Duval singularities.”

Let $\mathbb{A}^{3}$ be complex affine 3-space, and let $\textbf{A}_{1}\subset \mathbb{A}^{3}$ be the zero set of

$xz - y^{2}$

It is a hypersurface (2-dimensional over $\mathbb{C}$) with singularity at $\textbf{0}:=(0,0,0)$. Geometrically, its real part looks like two cones symmetric about the origin, joining at a singular point. The cones are a union of “generating lines” which all meet at $\textbf{0}$. The “blow up'” of $\textbf{A}_{1}$ at $\textbf{0}$ is simply a cylinder, whose underlying set is the disjoint union of these lines; thus the point $\textbf{0}$, in the real case, is replaced by a circle, called the exceptional divisor of the blow up. Over the complex numbers, the exceptional divisor is a copy of $\mathbb{P}^{1}$. The resulting curve is non-singular, so we have “resolved the singularity” at $\textbf{0}$. During this post, the exceptional divisor will always turn out to be the union of $\mathbb{P}^{1}$‘s which intersect in a finite number of points. In this case, the intersection diagram of the exceptional divisor is defined to be the planar graph whose points are the copies of $\mathbb{P}^{1}$, and whose edges correspond to the intersections. Therefore the diagram in the case of $\textbf{A}_{1}$ at $\textbf{0}$ is a single vertex.

As we’ve just seen, the idea of blowing up is to “pull apart” the multiple tangent lines at a singular point; if the resulting algebraic set is still singular, we may have to blow up more times to resolve the singularities. Here’s a more formal definition. There is a natural rational map $\phi: \mathbb{A}^{n}\to \mathbb{P}^{n-1}$ which sends $\mathbf{x}=(x_{1},\dots, x_{n})$ to ${}[\mathbf{x}]=[x_{1}:\cdots : x_{n}]$ and is defined everywhere but the origin. Thinking of $\mathbb{P}^{n-1}$ as the space of lines through $\mathbf{0}$ in $\mathbb{A}^n$, then $\phi(\mathbf{x})$ is the line from $\mathbf{x}$ to $\mathbf{0}$ when $\mathbf{x}\neq \mathbf{0}$. The blow up of $X\subset \mathbb{A}^n$ will be essentially the graph of $\phi$ restricted to $X$; as parts of $X$ approach the singularity at $\mathbf{0}$ along different lines, they will no longer collide in the graph, thus pulling apart the tangent lines (skip ahead to picture).

Formally, the graph of $\phi$ is defined to be the subspace of $\mathbb{A}^{n}\times \mathbb{P}^{n-1}$ with two pieces: the first consisting of all elements $(\mathbf{x},\phi(\mathbf{x})),$ $\mathbf{x}\neq \mathbf{0} \in X,$ which we call $\widetilde{X}$, and the second just $\{\textbf{0}\}\times \mathbb{P}^{n-1}$ (call it $\Pi$). So we see that the graph projects isomorphically to $X$ everywhere away from the origin, just by restricting to the first coordinate, and above the origin there is a whole copy of $\mathbb{P}^{n-1}$. Then the blow up $X_{bl}$ is defined as the Zariski closure of the first part $\widetilde{X}$ (the Zariski closure of a subset $Y\subset \mathbb{A}^n\times \mathbb{P}^{n-1}$ is just the smallest algebraic set containing $Y$). The subset of $X_{bl}$ which projects to the origin in $X$ is called the exceptional divisor; it is exactly $X_{bl}\cap \Pi$.

Here are two pictures to clarify this, both stolen shamelessly from Harris’s introduction (a great book!). The first shows the blow up of $\mathbb{A}^2$, (over the reals). From the picture one can see that every point downstairs $\neq \mathbf{0}$  corresponds to one point in the blow up, and points in the same line through $\mathbf{0}$ downstairs are at the same height upstairs. The line over the origin is a copy of $\mathbb{RP}^1$. The second picture is the blow up of the nodal cubic, which is just a subset of the blow up of $\mathbb{A}^2$. One can see how the two different tangent directions at $\mathbf{0}$ are pulled apart into a non-singular curve.

Given $(x_{1},\dots, x_{n})\in \mathbb{A}^{n}$, an element ${}[w_{1}:\cdots :w_{n}]\in \mathbb{P}^{n-1}$ is equal to ${}[x_{1}:\cdots : x_{n}]$ if and only if $x_{i}w_{j}=x_{j}w_{i}$ for all $i,j$. So another definition of the graph of $\phi$ restricted to $X$ (i.e. $\widetilde{X} \cup \Pi$) is as the set

$\{\left((x_{1},\dots x_{n}),{}[w_{1}:\cdots :w_{n}]\right)\in \mathbb{A}^{n}\times \mathbb{P}^{n-1}: (x_{1},\dots x_{n})\in X, x_{i}w_{j}=x_{j}w_{i}\}$

Using this description, we can actually compute the blow up of some surfaces. For example, let’s go back to

$\textbf{A}_{1}=\{(x,y,z)\in \mathbb{A}^{3}: xz - y^{2}=0\}$

Suppose we write an element of $\mathbb{P}^{2}$ as ${}[a:b:c]$. Then points of the blow up satisfy

$xb=ya$

$yc = zb$

$xc = za$

$xz - y^{2}=0$

Restricting, say, to $c=1$ corresponds to restricting to an affine chart in $\mathbb{P}^{2}$. In this case we combine the above equations to get

$z^{2}(b-a^{2})=0$

The first irreducible piece $z^{2}=0$ corresponds to $\Pi$, which is a copy of $\mathbb{P}^{2}$ lying above $\textbf{0}$ (as it should be). Taking the Zariski closure of the graph minus $\Pi$ thus corresponds to restricting to the other piece $b-a^{2}=0$, which is a smooth surface in $\mathbb{A}^{3}$ (and one can check that the blow up is smooth in the other charts as well). Remaining in the $c=1$ chart, restricting to $z=0$ gives the exceptional divisor, which is a curve $b-a^{2}=0$ in $\mathbb{A}^{2}$, and by degree arguments (e.g. it’s a hyperelliptic curve of degree 1 in $b$, so its genus is 0 by the Riemann Hurwitz theorem) it must be the affine part of a copy of $\mathbb{P}^{1}$, as we claimed before.

The surface $\textbf{A}_{1}$ also has another realization as a quotient variety. Namely, the group $\mathbb{Z}_{2}$ acts on $\mathbb{A}^{2}=\{( u,v)\}$ via $(u,v)\mapsto (-u,-v)$, and the quotient of $\mathbb{A}^{2}$ by this action gives a space which is a non-singular algebraic set everywhere except at the origin, where it is singular. The map $\mathbb{A}^{2}\to \mathbb{A}^{3}$ sending $(u,v)\mapsto (u^{2}, uv,v^{2})$ descends to this quotient variety, defining an isomorphism between the quotient variety and $\textbf{A}_{1}$.

In this way, any discrete subgroup of $GL(2,\mathbb{C})$ whose only fixed point is the origin $\textbf{0}$ will define a quotient variety which is non-singular except possibly at $\textbf{0}$. For example, the binary dihedral group $BD_{8}\subset GL(2,\mathbb{C})$ of order 8, which is a lift of a dihedral group $D_{8}\subset SO(3)$ to $SU(2)$, defines an action on $\mathbb{A}^{2}$ generated by the two maps

$\alpha : (u,v)\mapsto (iu,-iv)~~~\beta: (u,v)\mapsto (v,-u)$

Then the map from $\mathbb{A}^{2}\to \mathbb{A}^{3}$ sending

$(u,v)\mapsto ((u^{4}-v^{4})uv , u^{4}+v^{4},u^{2}v^{2})=(x,y,z)$

presents this quotient variety, which we call $\textbf{D}_{4}$, as a hypersurface in $\mathbb{A}^{3}$; we can change coordinates so that the hypersurface is the zero set of

$x^{2}+y^{3}+z^{3}$

Now, let’s blow up the singularity at $\textbf{0}$ for $\textbf{D}_{4}$ as we did for $\textbf{A}_{1}$. As before, we take the chart $c=1$ and get the equations

$x^{2}+y^{3}+z^{3}=0$

$ya = xb$

$za = xc = x$

$zb = cy = y$

Substituting, we obtain

$z^{2}(a^{2}+b^{3}z +z)=0$

so the blow up in this chart is just the (singular) surface

$a^{2}+z(b^{3} +1)=0$

after we discard $\Pi$. The exceptional divisor corresponds to $z=0$, whence $a=0$, so it is just the $b$ axis, giving a copy of $\mathbb{P}^{1}$. Call this $\Gamma_{0}$. In the other charts, we get

$a^{2} + y(c^{3}+1)=0$

and

$1+x(c^{3}+b^{3})=0$

From these equations we can see that all singularities were present in the first chart, at points $a=z=0, b^{3}+1=0$. At each of these points, we can change coordinates so that the singularity looks like the singularity in $\textbf{A}_{1}$. So we just have to blow up each of these points, and we will get something smooth. There is evidently a symmetry of the surface interchanging them, so, to compute the entire intersection diagram, it suffices to consider what happens at any one of them.

We take a fresh set of coordinates $x,y,z$, and blow up the curve

$x^{2}+z(y^{3} +1)=0$

at some singularity. To do this, take $\zeta^{3}=-1$, and define $y_{1}=y-\zeta$; then our equation becomes

$x^{2}+z(y_{1}^{3}+y_{1}^{2}\zeta+y_{1}\zeta^{2})=0$

which is singular at $\textbf{0}$. Blowing up as before and again taking the $c=1$ chart, we get

$a^{2}+z^{2}b^{3}+\zeta zb^{2}+\zeta^{2}b=0$

The exceptional divisor $\Gamma_{1}$ is the intersection of this surface with the plane $z=0$, which is $a^{2}+\zeta^{2}b=0$, again a copy of $\mathbb{P}^{1}$. Blowing up $\Gamma_{0}$ in this chart, we get the $b$ axis, i.e., $a=z=0$. So the intersection of the blow up of $\Gamma_{0}$ with $\Gamma_{1}$ is the point $a=z=b=0$. By symmetry, the other exceptional divisors (corresponding to the other third roots of -1) intersect $\Gamma_{0}$ in the same way, so our final intersection diagram is exactly

with $\Gamma_{0}$ in the middle. This is the Dynkin diagram corresponding to the lie algebra $D_{4}$! Of course, the intersection diagram for $\textbf{A}_{1}$ was the dynkin diagram $A_{1}$. Thus we have a correspondance of groups and diagrams, $\mathbb{Z}_{2}\mapsto A_{1}$, $BD_{8}\mapsto D_{4}$. In fact, this completely generalizes. Here is Reid’s table of all the Duval singularities:

And here, as promised, is Arnol’d’s:

Play around with this for a while! The table has an exposition in Arnol’d’s book, here (.djvu)

Tamar has generalized this up a dimension or two; she found that the same thing works as long as you DEFINE AN ENTIRELY NEW LIE ALGEBRA-LIKE STRUCTURE! She calls them Lie algebras of the third kind (LATKe’s). Here’s her paper.