On rational homology disk smoothings of valency $4$ surface singularities

Abstract

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 (“$\mathbb{Q}HD$”) 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 $\mathbb{Q}HD$ smoothings for those classes; it is an explicit $\mathbb{Q}$–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 $\mathbb{Q}HD$ smoothings for the first class.

We also prove a general formula for the dimension of a $\mathbb{Q}HD$ 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 $\mathbb{Q}HD$ smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a general conjecture:

Conjecture The only complex surface singularities admitting a $\mathbb{Q}HD$ smoothing are the (known) weighted homogeneous examples.