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Complex surface singularities with rational homology disk smoothings
Algebraic Geometry · Submitted 25 Jun 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 $0$) ([9], 5.9.1). In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szab\'{o}, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. In his UNC PhD thesis (unpublished but available at [2]), our student Jacob Fowler completed the analytic classification of these singularities, and counted the number of smoothings in each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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $0$) ([9], 5.9.1). | background 1.00 | 0.05 | 0.05 | asserted 1.00 |
| s1 | In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. | background 0.68 | 0.91 | 0.03 | asserted 1.00 |
| s2 | Later work of Stipsicz, Szab\'{o}, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. | background 0.99 | 0.10 | 0.04 | asserted 1.00 |
| s3 | In his UNC PhD thesis (unpublished but available at [2]), our student Jacob Fowler completed the analytic classification of these singularities, and counted the number of smoothings in each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 $\mathcal M$. | gap 0.86 | 0.12 | 0.04 | asserted 1.00 |
| s4 | 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. | contribution 0.59 | 0.88 | 0.88 | asserted 1.00 |
| s5 | The method involves study of configurations of rational curves on projective rational surfaces. | method 1.00 | 0.55 | 0.03 | asserted 0.99 |
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Original 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 $0$) ([9], 5.9.1). In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szab\'{o}, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. In his UNC PhD thesis (unpublished but available at [2]), 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.
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2 arXiv papers cite this in 5 sentences
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Cites it as support 1 paper
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According to Wahl (cf. [ref] , Theorem 1.8), there is a unique smoothing component, so such a Stein filling $W$ is uniquely determined.
Small symplectic $4$-manifolds via contact gluing and some applications · Introduction and the main results · 0.98
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