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  "body": "\n## Description\n\nLet $k$ be a field and let $X \\subset \\mathbb{P}^n_k$ be a closed subscheme of [[projective space|ZT9A6JK8QF]], for example a [[projective variety|A4H4CZ2V35]].\nThe *Hilbert functor* of $X$ sends a $k$-scheme $T$ to the set of closed subschemes $Z \\subset X \\times_k T$ such that the projection $Z \\to T$ is [[flat|5M6KS8F47R]] and [[proper|9537RKBZPP]].\nGrothendieck proved that this functor is [[representable|KCCHH87H6E]].\nSo there are a $k$-scheme $\\operatorname{Hilb}(X)$ and a universal flat family $\\mathcal{Z} \\subset X \\times \\operatorname{Hilb}(X)$.\nEvery family $Z \\subset X \\times T$ as above is the pullback of $\\mathcal{Z}$ along a unique morphism $T \\to \\operatorname{Hilb}(X)$.\nThis is the *Hilbert scheme* of $X$.\nIt is a disjoint union $\\operatorname{Hilb}(X) = \\coprod_P \\operatorname{Hilb}^P(X)$ over the [[Hilbert polynomials|5FR6JX04RS]] $P$ of the fibers, and each $\\operatorname{Hilb}^P(X)$ is a projective scheme.\nSo the Hilbert scheme is a fine [[moduli space|FJZRW9P0K3]] for closed subschemes of $X$, in the sense of the [[functor of points|SWWW9X4A3N]].\n\n## Remarks\n\nThe hypersurfaces of degree $d$ in $\\mathbb{P}^n$ are the zero sets of nonzero homogeneous polynomials of degree $d$ up to scalars.\nTheir Hilbert scheme is the projective space $\\mathbb{P}^N$ with $N = \\binom{n+d}{d} - 1$, whose points are these polynomials up to scaling.\nThe Hilbert scheme also exists for quasi-projective $X$, with quasi-projective components.\nThe Hilbert scheme of $m$ points in the plane, $\\operatorname{Hilb}^m(\\mathbb{A}^2)$, is smooth and irreducible of dimension $2m$ by a theorem of Fogarty.\nIt resolves the singularities of the symmetric product of $m$ copies of $\\mathbb{A}^2$.\n\n## Explanations\n\n### L1 human\n\nThe Hilbert scheme is a space whose points are the subvarieties of a fixed projective variety, together with their degenerate limits.\nFamilies of subvarieties become curves or surfaces inside this space.\nFor example, the lines in $\\mathbb{P}^3$ form a four-dimensional Hilbert scheme, the Grassmannian of lines.\nIt is one of the first and most important examples of a moduli space.\n",
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