The number of equisingular moduli of a rational surface singularity

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).