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  "body": "\n## Summary\n\nA point of an algebraic variety is singular when the tangent space there is larger than the dimension of the variety.\nOtherwise it is nonsingular, or smooth.\nFor a hypersurface $f = 0$ with $f$ irreducible, the singular points are those where $f$ and all its partial derivatives vanish.\nSingular points are where the variety fails to look like a manifold, such as cusps and self-crossings.\nThey form a proper closed subset of every variety.\n\n## Explanations\n\n### L1 human\n\nMost points of a curve look like a small piece of a line when you zoom in.\nA singular point is one that does not: the curve crosses itself there, or has a sharp cusp.\nThe curve $y^2 = x^2 (x + 1)$ crosses itself at the origin, and the curve $y^2 = x^3$ has a cusp there.\nAt those points the partial derivatives of the equation both vanish, and there is no well-defined tangent line.\n",
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