Durfee’s conjecture on the signature of smoothings of surface singularities. (La conjecture de Durfee sur la signature des lissages des singularités des surfaces.) (English. French summary) Zbl 1382.32020
The authors prove some inequalities between invariants of a normal surface singularity related to A. H. Durfee conjectured inequalities given in [Math. Ann. 232, 85–98 (1978; Zbl 0346.32016)]. The main theorem is: Let \((X,0)\) be a normal Gorenstein surface singularity with embedding dimension \(e\) and geometric genus \(p_{g}.\) Let \(\sigma \) denote the signature of a smoothing of \((X,0)\) (it is the signature of the intersection form in \(H_{2}(F,\mathbb{Z)}\), where \(F\) is a generic fiber of a smoothing of \((X,0)\)). Then
1. If the resolution intersection form is unimodular, then \(-\sigma \geq 2^{4-e}(p_{g}+1).\)
2. If \((X,0)\) is a hypersurface singularity, then \(-\sigma \geq p_{g}+s_{\min},\) where \(s_{\min }\) is the number of irreducible exceptional curves in the minimal resolution of \(X.\)
1. If the resolution intersection form is unimodular, then \(-\sigma \geq 2^{4-e}(p_{g}+1).\)
2. If \((X,0)\) is a hypersurface singularity, then \(-\sigma \geq p_{g}+s_{\min},\) where \(s_{\min }\) is the number of irreducible exceptional curves in the minimal resolution of \(X.\)
Reviewer: Tadeusz Krasiński (Łódź)
MSC:
| 32S05 | Local complex singularities |
| 32S25 | Complex surface and hypersurface singularities |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
