 Singular point of an algebraic variety

A point $p$ of an irreducible variety $X$ is singular when $\dim T_{p}X>\dim X$. Equivalently, its local ring is not regular. The singular locus is the complement of the smooth locus.

A singular point of an algebraic variety is a point where the variety is not well-behaved in terms of its geometric structure. More formally, a point \( P \) on an algebraic variety \( V \) defined by a set of polynomial equations is termed a singular point if the local behavior of the variety at that point exhibits some form of "singularity," meaning that it fails to meet certain smoothness conditions.