A surface singularity. Let \(k\) be an algebraically closed field, and let \(X\)
be the surface in \(\mathbf{A}_{k}^{3}\) defined by the equation
\(x^{2}+y^{3}+z^{5}=0 .\) It has an isolated singularity at the origin
\(P=(0,0,0)\)
(a) Show that the affine ring \(A=k[x, y, z] /\left(x^{2}+y^{3}+z^{5}\right)\)
of \(X\) is a unique factorization domain, as follows. Let \(t=z^{-1} ; u=t^{3}
x,\) and \(v=t^{2} y .\) Show that \(z\) is irreducible in \(A ; t \in k[u, v],\) and
\(A\left[z^{-1}\right]=k\left[u, v, t^{-1}\right] .\) Conclude that \(A\) is a
UFD.
(b) Show that the singularity at \(P\) can be resolved by eight successive
blowings-up. If \(\tilde{X}\) is the resulting nonsingular surface, then the
inverse image of \(P\) is a union of eight projective lines, which intersect
each other according to the Dynkin \(\operatorname{diagram}
\mathbf{E}_{\mathbf{s}}\).
Here each circle denotes a line, and two circles are joined by a line segment
whenever the corresponding lines intersect. Note. This singularity has
interesting connections with local algebra, invariant theory, and topology.
In case \(k=\mathbf{C},\) Mumford [6] showed that the completion \(\hat{A}\) of
the ring \(A\) at the maximal ideal \(\mathrm{m}=(x, y, z)\) is also a UFD. This
is remarkable, because in general the completion of a local UFD need not be
UFD, although the converse is true (theorem of Mori) - see Samuel \([3] .\)
Brieskorn [2] showed that the corresponding analytic local ring
\(\mathbf{C}\\{x, y, z\\} /\left(x^{2}+y^{3}+z^{5}\right)\) is the only
nonregular normal 2 -dimensional analytic local ring which is a UFD. Lipman
[2] generalized this as follows: over any algebraically closed field \(k\) of
characteristic \(\neq 2,3,5,\) the only nonregular normal complete 2
-dimensional local ring which is a UFD is \(k[[x, y, z]]
/\left(x^{2}+y^{3}+z^{5}\right)\) See also Lipman [3] for a report on recent
work connected with UFD's.
This singularity arose classically out of Klein's work on the icosahedron. The
group \(I\) of rotations of the icosahedron, which is isomorphic to the simple
group of order \(60,\) acts naturally on the 2 -sphere. Identifying the 2
-sphere with \(\mathbf{P}_{\mathrm{C}}^{1}\) by stereographic projection, the
group \(I\) appears as a finite subgroup of Aut \(\mathbf{P}_{\mathbf{C}}^{1}\).
This action lifts to give an action of the binary icosahedral group \(\bar{I}\)
on \(\mathbf{C}^{2}\) by linear transformations of the complex variables \(t_{1}\)
and \(t_{2} .\) Klein \([2, \mathrm{I}, 2,813, \text { p.62 }]\) found three
invariant polynomials \(x, y, z\) in \(t_{1}\) and \(t_{2},\) related by the
equation \(x^{2}+y^{3}+\) \(z^{5}=0 .\) Thus the surface \(X\) appears as the
quotient of \(\mathbf{A}_{\mathbf{C}}^{2}\) by the action of the group
\(\bar{I}\). In particular, the local fundamental group of \(X\) at \(P\) is just
\(\bar{I}\).
With regard to the topology of algebraic varieties over \(C,\) Mumford [6]
showed that a normal algebraic surface over \(\mathbf{C}\), whose underlying
topological space (in its "usual" topology) is a topological manifold, must be
nonsingular. Brieskorn showed that this is not so in higher dimensions. For
example, the underlying topological space of the hypersurface in
\(\mathbf{C}^{4}\) defined by \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{3}=0\) is a
manifold. Later Brieskorn [1] showed that if one intersects such a singularity
with a small sphere around the singular point, then one may get a topological
sphere whose differentiable structure is not the standard one. Thus for
example, by intersecting the singularity
\\[
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{3}+x_{5}^{6 k-1}=0
\\]
in \(\mathbf{C}^{5}\) with a small sphere around the origin, for \(k=1,2, \ldots,
28,\) one obtains all 28 possible differentiable structures on the 7 -sphere.
See Hirzebruch and Mayer [1] for an account of this work.
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