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  "body": "\n## Summary\n\nA flat morphism of schemes is a morphism whose local rings are flat modules over the local rings of the target.\nFlatness is the algebraic expression of a family whose fibres vary continuously.\nIn a flat family of projective schemes over a connected base, numerical invariants such as the Hilbert polynomial stay constant from fibre to fibre.\n\n## Explanations\n\n### L1 human\n\nA flat morphism is a family of spaces with no sudden jumps.\nOpen immersions and the projection $\\mathbb{A}^2 \\to \\mathbb{A}^1$ are flat.\nThe blow-up of the plane at a point is not flat: all its fibres are points except one, which is a whole line.\nFlatness forbids such a jump in the fibre dimension, and also subtler jumps, such as a fibre suddenly gaining an embedded point.\n",
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