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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] and let $A$ be an [[analytic subset|VMDTGTMN9P]] of $X$.\nA point $a \\in A$ is a *regular point* of $A$ if it has an open neighbourhood $U$ in $X$ with the following property.\nThe set $A \\cap U$ is a complex [[submanifold|VKFW27WXEZ]] of $U$.\nA point $a \\in A$ is a *singular point* of $A$ if it is not a regular point.\nThe sets of regular and singular points are written $\\operatorname{reg} A$ and $\\operatorname{sing} A$.\n\n## Remarks\n\nExample: for the cusp $A = \\set{z_1^2 = z_2^3}$ in $\\C^2$, every point $a \\ne 0$ is regular, since the gradient $(2 z_1, -3 z_2^2)$ of $f = z_1^2 - z_2^3$ is nonzero there and the holomorphic implicit function theorem applies.\nThe origin is singular.\nIf $A$ were a complex submanifold near $0$, it would have dimension $1$ and be a graph over its tangent line at $0$.\nThe secant directions from $0$ to $(t^3, t^2)$ tend to the $z_2$-axis, so this tangent line is the $z_2$-axis, but the projection $(t^3, t^2) \\mapsto t^2$ to it is two-to-one near $0$.\nRegularity is a property of the set, not of the equations.\nThe set $A = \\set{z_1^2 = 0}$ in $\\C^2$ is the line $\\set{z_1 = 0}$, so all its points are regular, although the gradient of $z_1^2$ vanishes on $A$.\nThe set $\\operatorname{reg} A$ is open in $A$ by definition, and it is dense in $A$.\nThe set $\\operatorname{sing} A$ is a closed analytic subset of $X$ that is nowhere dense in $A$; this is a theorem, not a direct consequence of the definition.\n",
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