The number of equisingular moduli of a rational surface singularity

TL;DR

This paper investigates a topological inequality related to the number of equisingular moduli in rational surface singularities, proving it in special cases and establishing a classification in characteristic p.

Contribution

It proves the conjectured inequality in certain cases and extends the classification of taut singularities in characteristic p.

Findings

Verified the inequality when the resolution dual graph is sufficiently negative.

Established a more general proof in characteristic p using a hard vanishing theorem.

Classified all taut singularities with reduced fundamental cycle in characteristic p.

Abstract

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).