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  "body": "\n## Summary\n\nA point of a [[complex analytic variety|GB7M582M5Y]] is regular when the set is a complex submanifold near that point, and singular otherwise.\nThe regular points form an open dense subset of the variety, and the singular points form a nowhere dense analytic subset.\nSingular points are where the local geometry is interesting, and their study is the subject of singularity theory.\n\n## Explanations\n\n### L1 human\n\nNear a regular point, an analytic set looks like a flat piece of $\\C^d$, bent smoothly.\nAt a singular point something goes wrong: several pieces cross, or the set has a sharp tip.\nThe curve $\\set{z_1 z_2 = 0}$ is singular at the origin, where two lines cross.\nThe cusp $\\set{z_1^2 = z_2^3}$ is a single piece, but it is singular at the origin, where it has a sharp point.\nAll other points of these two curves are regular.\n",
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