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Thanks to recent work of Stipsicz, Szabó and the author and of Bhupal and
Stipsicz, one has a complete list of resolution graphs of weighted homogeneous
complex surface singularities admitting a rational homology disk (“QHD”)
smoothing, that is, one with Milnor number 0. They fall into several classes, the most
interesting of which are the 3 classes whose resolution dual graph has central
vertex with valency 4. We give a uniform “quotient construction” of the QHD
smoothings for those classes; it is an explicit Q–Gorenstein smoothing, yielding
a precise description of the Milnor fibre and its non-abelian fundamental
group. This had already been done for two of these classes; what is new
here is the construction of the third class, which is far more difficult. In
addition, we explain the existence of two different QHD smoothings for the first
class.
We also prove a general formula for the dimension of a QHD smoothing
component for a rational surface singularity. A corollary is that for the valency 4
cases, such a component has dimension 1 and is smooth. Another corollary is that
“most” H–shaped resolution graphs cannot be the graph of a singularity with a QHD
smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a
general conjecture:
Conjecture The only complex surface singularities admitting a QHD
smoothing are the (known) weighted homogeneous examples.
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