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A point $p$ on a quasi-projective variety $V$ is called smooth if $$dimT_p(V)=dim_p(V)$$ otherwise $p\in V$ is singular. Could any one explain me with an example of a singular point and how their locus forms a closed subset of $V$?

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Take for example $V=Z(y^2-x^3) \subseteq \mathbb{A}^2$. This is a quasiprojective variety, and by the Jacobian criterion (for example), every point on V $\neq (0,0)$ is non-singular. However, the tangent space at the origin is 2-dimensional, whereas the variety is of dimension 1.

Draw V. Think of the tangent space at the origin as a double line.

Since the singular locus consists of only one point, it is a closed subset of V.

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