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
(“”) smoothing, that is,
one with Milnor number .
They fall into several classes, the most interesting of which are the
classes whose resolution dual graph has central vertex with valency
.
We give a uniform “quotient construction” of the
smoothings for those classes;
it is an explicit –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
smoothings for the first class.
We also prove a general formula for the dimension of a
smoothing
component for a rational surface singularity. A corollary is that for the valency
cases, such a component
has dimension
and is smooth. Another corollary is that “most”
–shaped
resolution graphs cannot be the graph of a singularity with a
smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a
general conjecture:
Conjecture The only complex surface singularities admitting asmoothing are the (known) weighted homogeneous examples.
