Durfee's conjecture on the signature of smoothings of surface singularities

Kollár, J. and Némethi, András and De Fernex, T. (2017) Durfee's conjecture on the signature of smoothings of surface singularities. ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE, 50 (3). pp. 787-798. ISSN 0012-9593

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Abstract

In 1978 Durfee conjectured various inequalities between the signature σ and the geometric genus pg of a normal surface singularity. Since then a few counter examples have been found and positive results established in some special cases. We prove a 'strong' Durfee-type inequality for any smoothing of a Gorenstein singularity, provided that the intersection form of the resolution is unimodular. We also prove the conjectured 'weak' in- equality for all hypersurface singularities and for sufficiently large multiplicity strict complete intersec- tions. The proofs establish general inequalities valid for any numerically Gorenstein normal surface singularity. © 2017 Société Mathématique de France. Tous droits réservés.

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