Methods and Applications of Analysis

Volume 24 (2017)

Number 1

Special issue dedicated to Henry B. Laufer on the occasion of his 70th birthday: Part 1

Guest Editors: Stephen S.-T. Yau (Tsinghua University, China); Gert-Martin Greuel (University of Kaiserslautern, Germany); Jonathan Wahl (University of North Carolina, USA); Rong Du (East China Normal University, China); Yun Gao (Shanghai Jiao Tong University, China); and Huaiqing Zuo (Tsinghua University, China)

The number of equisingular moduli of a rational surface singularity

Pages: 125 – 153



Jonathan Wahl (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.)


We consider a conjectured topological inequality for the number of equisingular moduli of a rational surface singularity, and prove it in some natural special cases. When the resolution dual graph is “sufficiently negative” (in a precise sense), we verify the inequality via an easy cohomological vanishing theorem, which implies that this number is computed simply from the graph (Theorem 3.10). To consider an important and less restrictive meaning of “sufficiently negative” requires a much more difficult “hard vanishing theorem” (Theorem 4.5), which is false in characteristic $p$. Theorem 7.9 verifies the conjectured inequality in this more general situation. As a corollary, we classify in characteristic $p$ all taut singularities with reduced fundamental cycle (Theorem 9.2).


rational singularity, equisingular deformation, tautness, characteristic $p$ singularities, quasi-homogeneity

2010 Mathematics Subject Classification

14B07, 14J17, 32S15, 32S25

Received 27 March 2016

Published 18 August 2017