nLab singular point of an algebraic variety

Idea

An algebraic variety $X \hookrightarrow \mathbb{A}^n$ defined as the zero locus of polynomials $\{f^i \in R[x_1, \cdots, x_n]\}$ is singular at a point $p$, if the Jacobian matrix of first derivatives at this point has rank lower than at other points.

A variety is called singular if it has at least one singular point and non-singular if it has none.

Examples

• cusp

• nodal curve

Related concepts

• geometric point

References

• Wikipedia, Singular point of an algebraic variety