---
tag: D86X15A8WS
kind: concept
name: Singular point of an algebraic variety
msc: [14B05, 14A10]
sources:
  - url: https://en.wikipedia.org/wiki/Singular_point_of_an_algebraic_variety
same_as:
  wikidata: Q7524253
  wikipedia: Singular point of an algebraic variety
---

## Summary

A point of an algebraic variety is singular when the tangent space there is larger than the dimension of the variety.
Otherwise it is nonsingular, or smooth.
For a hypersurface $f = 0$ with $f$ irreducible, the singular points are those where $f$ and all its partial derivatives vanish.
Singular points are where the variety fails to look like a manifold, such as cusps and self-crossings.
They form a proper closed subset of every variety.

## Explanations

### L1 human

Most points of a curve look like a small piece of a line when you zoom in.
A singular point is one that does not: the curve crosses itself there, or has a sharp cusp.
The curve $y^2 = x^2 (x + 1)$ crosses itself at the origin, and the curve $y^2 = x^3$ has a cusp there.
At those points the partial derivatives of the equation both vanish, and there is no well-defined tangent line.