---
tag: 98QASBAMEB
kind: concept
name: Rational singularity
msc: [14B05, "14E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_singularity
same_as:
  wikidata: Q7295783
  wikipedia: Rational singularity
---

## Summary

A rational singularity is a singular point that looks, from the point of view of sheaf cohomology, like a smooth point.
It is defined by the vanishing of the higher direct images of the structure sheaf under a [[resolution|MB52NDA6WN]].
Quotient singularities, toric singularities and the [[ADE singularities|HQNXQBZ6C7]] are rational.
Rational singularities play a central role in the [[minimal model program|Q8CTDCZH01]], since the log terminal singularities it uses are rational.

## Explanations

### L1 human

When a singular point is resolved, the new smooth variety may carry extra cohomology above the point.
A singularity is rational when no such extra cohomology appears in the structure sheaf.
The tip of the cone $x^2 + y^2 = z^2$ is rational: its resolution adds a single projective line, which has no first cohomology.
The tip of the cone over a smooth plane cubic is not: its resolution adds an elliptic curve, whose first cohomology is one-dimensional.