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  "body": "\n## Summary\n\nThe dimension of a [[complex analytic variety|GB7M582M5Y]] at a point measures how many complex parameters are needed to describe it near that point.\nAt a [[regular point|ZW5A88PCPK]] it is the dimension of the set as a complex manifold.\nAt a singular point it is the largest dimension of the manifold pieces nearby.\nIt agrees with the Krull dimension of the local ring of the set, and with the number $d$ of the local parametrization theorem.\n\n## Explanations\n\n### L1 human\n\nA curve has dimension $1$ and a surface dimension $2$, counted in complex parameters.\nAn analytic set can mix pieces of different dimensions.\nIn $\\C^3$, take the union of the plane $\\set{z_3 = 0}$ and the line $\\set{z_1 = z_2 = 0}$.\nIt has dimension $2$ at the origin, where the plane passes, and dimension $1$ at the points of the line away from the plane.\n",
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