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  "body": "\n## Definition\n\nLet $X$ be an [[algebraic variety|HV6YBS0YDH]] over an algebraically closed field $k$ and let $p \\in X$.\nThe point $p$ is *nonsingular* (or smooth) if $\\dim_k T_p X = \\dim X$, where $T_p X$ is the [[Zariski tangent space|E7A5DHFT01]] and $\\dim X$ is the [[dimension|36X2WTHYCB]] of $X$.\nOtherwise $\\dim_k T_p X \u003e \\dim X$, and $p$ is a *singular point* of $X$.\nThe variety $X$ is *nonsingular* if it has no singular point.\n\n## Remarks\n\nJacobian criterion: let $X \\subseteq k^n$ be an [[affine variety|JERVQT4C12]] of dimension $d$ with $I(X) = (f_1, \\dots, f_r)$.\nThen $p$ is nonsingular if and only if the [[Jacobian matrix|047NDTEPCV]] $\\left(\\frac{\\partial f_j}{\\partial x_i}(p)\\right)$ has rank $n - d$.\nAlgebraic form: $p$ is nonsingular if and only if the [[local ring|MZ02BKPHJA]] of $X$ at $p$ is a [[regular local ring|63XXTF6B5M]].\nThe singular points of $X$ form a closed subset different from $X$.\n\nExample: for $X = V(f) \\subseteq k^2$ with $f = y^2 - x^2(x + 1)$, a curve of dimension $1$, the [[partial derivatives|09EGC6F8PT]] are $\\partial f / \\partial x = -3x^2 - 2x$ and $\\partial f / \\partial y = 2y$.\nAssume $\\operatorname{char} k \\ne 2$.\nBoth vanish together with $f$ only at $(0, 0)$, so the origin is the only singular point.\n\nNon-example: on the circle $V(x^2 + y^2 - 1)$ with $\\operatorname{char} k \\ne 2$, the gradient $(2x, 2y)$ vanishes only at $(0, 0)$, which is not on the circle, so the circle has no singular point.\n",
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