Complex Surface Singularities with Rational Homology Disk Smoothings

Abstract

A cyclic quotient singularity of type p ²∕pq − 1 (0 < q < p, (p, q) = 1)) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is 0). In the 1980s, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szabó, Bhupal, and the author proved that these were the only weighted homogeneous examples. In his UNC PhD thesis, our student Jacob Fowler completed the analytic classification of these singularities, and counted the number of smoothings in each case, except for types $\mathcal W$, $\mathcal N$, and $\mathcal M$. In this paper, we describe his results, and settle these remaining cases; there is a unique $\mathbb Q$HD smoothing component except in the cases of an obvious symmetry of the resolution dual graph. The method involves study of configurations of rational curves on projective rational surfaces.

To András Némethi on his 60th birthday

Appendix

See Tables A.1 and A.2.

Table A.1 Graphs in the families $\mathcal {W}, \mathcal {N}, \mathcal {M}, \mathcal {B}^3_2, \mathcal {C}^3_2 , \mathcal {C}^3_3 , \mathcal {A}^4, \mathcal {B}^4,\ \text{and}\ \mathcal {C}^4$

Table A.2 The dual graphs to the graphs in the families $\mathcal {W}, \mathcal {N}, \mathcal {M}, \mathcal {B}^3_2, \mathcal {C}^3_2, \mathcal {C}^3_3 , \mathcal {A}^4, \mathcal {B}^4,\ \text{and}\ \mathcal {C}^4$