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The weighted-homogeneity conjecture for rational surface singularities with rational homology disk smoothings

A rational surface singularity is a normal complex surface singularity whose resolution has a negative-definite exceptional divisor and whose geometric genus is zero. A rational homology disk smoothing (or Q\mathbb QHD smoothing) is a smoothing whose Milnor fibre has the rational homology of a disk.

Weighted-homogeneity conjecture. Suppose a rational surface singularity has a Q\mathbb QHD smoothing. Then the singularity is weighted homogeneous, hence is one of the known examples.

The conjecture is motivated by the classification of known examples and by results ruling out many resolution graphs, while the paper establishes dimension and smoothness results for Q\mathbb QHD smoothing components in valency 44 cases. It remains open in the stated generality.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The weighted-homogeneity conjecture for rational surface singularities with rational homology disk smoothings

    A rational surface singularity is a normal complex surface singularity with geometric genus zero. A rational homology disk smoothing (or Q\mathbb QHD smoothing) is a smoothing whose Milnor fibre has the rational homology of a disk.

    Weighted-homogeneity conjecture. Suppose a rational surface singularity has a Q\mathbb QHD smoothing. Then the singularity is weighted homogeneous, hence is one of the known examples.

    This is the detailed formulation of the same conjecture stated earlier in the paper. The discussion notes that the known quotient-construction examples are weighted homogeneous and that results on resolution graphs provide evidence, but the assertion is not proved in general.

    source: Jonathan Wahl, “On rational homology disk smoothings of valency 4 surface singularities”, arXiv:1005.2199 (2011).

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Primary source

Jonathan Wahl, “On rational homology disk smoothings of valency 4 surface singularities”, arXiv:1005.2199 (2011).

singularity-theorysmoothingssurface-singularities

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