  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).
