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  "body": "\n## Summary\n\nDeformation theory studies infinitesimal variations of a geometric or algebraic object: families over small base spaces whose special member is the given object.\nIn algebraic geometry the small base spaces are spectra of local Artinian rings, and the first step is the ring of dual numbers $k[\\varepsilon]/(\\varepsilon^2)$.\nFor a smooth variety, first-order deformations are classified by the first cohomology group of the tangent sheaf.\nThe obstructions to extending them lie in the second cohomology group.\nIt describes the local structure of [[moduli spaces|FJZRW9P0K3]] near a point.\n\n## Explanations\n\n### L1 human\n\nGiven a curve, one can ask in how many independent directions it can be bent while staying a curve of the same kind.\nDeformation theory answers this question to first order, like a derivative, and then asks whether each first-order bending extends to an actual family.\nThe projective line cannot be deformed at all: every small deformation is again a projective line.\nA smooth curve of genus $g \\ge 2$ has $3g - 3$ independent directions of deformation, which is the dimension of the moduli space of such curves.\n",
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