Hilbert scheme

 ( Scheme Theory ) 

In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme theory that is the parameter space for the of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.

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Hilbert scheme of projective space The Hilbert scheme $\backslash mathbf\{Hilb\}(n)$ of $\backslash mathbb\{P\}^n$ classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points

$\backslash operatorname\{Hom\}(S,\; \backslash mathbf\{Hilb\}(n))$

of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of $\backslash mathbb\{P\}^n\; \backslash times\; S$ that are flat morphism over . The closed subschemes of $\backslash mathbb\{P\}^n\; \backslash times\; S$ that are flat over can informally be thought of as the families of subschemes of projective space parameterized by . The Hilbert scheme $\backslash mathbf\{Hilb\}(n)$ breaks up as a disjoint union of pieces $\backslash mathbf\{Hilb\}(n,\; P)$ corresponding to the Hilbert scheme of the subschemes of projective space with Hilbert polynomial . Each of these pieces is projective over $\backslash operatorname\{Spec\}(\backslash Z)$.

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Construction as a determinantal variety Grothendieck constructed the Hilbert scheme $\backslash mathbf\{Hilb\}(n)$ of $n$-dimensional projective $\backslash mathbb\{P\}^n$ space as a subscheme of a Grassmannian defined by the vanishing of various . Its fundamental property is that for a scheme $T$, it represents the functor whose $T$-valued points are the closed subschemes of $\backslash mathbb\{P\}^n\; \backslash times\; T$ that are flat over $T$.

If $X$ is a subscheme of $n$-dimensional projective space, then $X$ corresponds to a graded ideal $I\_X^\backslash bullet$ of the polynomial ring $S$ in $n+1$ variables, with graded pieces $I\_X^m$. For sufficiently large $m$ all higher cohomology groups of $X$ with coefficients in $\backslash mathcal\{O\}(m)$ vanish. Using the exact sequence

$0\; \backslash to\; I\_X\; \backslash to\; \backslash mathcal\{O\}\_\{\backslash mathbb\{P\}^n\}\; \backslash to\; \backslash mathcal\{O\}\_X\; \backslash to\; 0$

we have $I\_X^m\; =\; \backslash Gamma(I\_X\backslash otimes\; \backslash mathcal\{O\}\_\{\backslash mathbb\{P\}^n\}(m))$ has dimension $Q(m)\; -\; P\_X(m)$, where $Q$ is the Hilbert polynomial of projective space. This can be shown by tensoring the exact sequence above by the locally flat sheaves $\backslash mathcal\{O\}\_\{\backslash mathbb\{P\}^n\}(m)$, giving an exact sequence where the latter two terms have trivial cohomology, implying the triviality of the higher cohomology of $I\_X(m)$. Note that we are using the equality of the Hilbert polynomial of a coherent sheaf with the Euler-characteristic of its sheaf cohomology groups.

Pick a sufficiently large value of $m$. The $(Q(m)\; -\; P\_X(m))$-dimensional space $I\_X^m$ is a subspace of the $Q(m)$-dimensional space $S^m$, so represents a point of the Grassmannian $\backslash textbf\{Gr\}(Q(m)-P\_X(m),\; Q(m))$. This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial $P\_X$ into this Grassmannian.

It remains to describe the scheme structure on this image, in other words to describe enough elements for the ideal corresponding to it. Enough such elements are given by the conditions that the map has rank at most for all positive , which is equivalent to the vanishing of various determinants. (A more careful analysis shows that it is enough just to take .)

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Universality Given a closed subscheme $Y\; \backslash subset\; \backslash mathbb\{P\}^n\_k=X$ over a field with Hilbert polynomial $P$, the Hilbert scheme has a universal subscheme $W\; \backslash subset\; X\; \backslash times\; H$ flat over $H$ such that

• The fibers $W\_x$ over closed points $x\; \backslash in\; H$ are closed subschemes of $X$. For $Y\; \backslash subset\; X$ denote this point $x$ as $Y\; \backslash in\; H$.
• $H$ is universal with respect to all flat families of subschemes of $X$ having Hilbert polynomial $P$. That is, given a scheme $T$ and a flat family $W\text{'}\; \backslash subset\; X\backslash times\; T$, there is a unique morphism $\backslash phi:\; T\; \backslash to\; H$ such that $\backslash phi^*W\; \backslash cong\; W\text{'}$.

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Tangent space The tangent space of the point $Y\; \backslash in\; H$ is given by the global sections of the normal bundle $N\_\{Y/X\}$; that is,

$T\_\{Y\}H\; =\; H^0(Y,\; N\_\{Y/X\})$

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Unobstructedness of complete intersections For local complete intersections $Y$ such that $H^1(Y,N\_\{X/Y\})\; =\; 0$, the point $Y\backslash in\; H$ is smooth. This implies every deformation of $Y$ in $X$ is unobstructed.

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Dimension of tangent space In the case $H^1(Y,N\_\{X/Y\})\; \backslash neq\; 0$, the dimension of $H$ at $Y$ is greater than or equal to $h^0(Y,N\_\{X/Y\})\; -\; h^1(Y,N\_\{X/Y\})$.

In addition to these properties, determined for which polynomials the Hilbert scheme $\backslash mathbf\{Hilb\}(n,\; P)$ is non-empty, and showed that if $\backslash mathbf\{Hilb\}(n,\; P)$ is non-empty then it is linearly connected. So two subschemes of projective space are in the same connected component of the Hilbert scheme if and only if they have the same Hilbert polynomial.

Hilbert schemes can have bad singularities, such as irreducible components that are non-reduced at all points. They can also have irreducible components of unexpectedly high dimension. For example, one might expect the Hilbert scheme of points (more precisely dimension 0, length subschemes) of a scheme of dimension to have dimension , but if its irreducible components can have much larger dimension.

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Functorial interpretation There is an alternative interpretation of the Hilbert scheme which leads to a generalization of relative Hilbert schemes parameterizing subschemes of a relative scheme. For a fixed base scheme $S$, let $X\; \backslash in\; (Sch/S)$ and let

$\backslash underline\{$

   \text{Hilb}
     

}_{X/S}:(Sch/S)^{op} \to Sets

be the functor sending a relative scheme $T\; \backslash to\; S$ to the set of isomorphism classes of the set

$\backslash underline\{$

   \text{Hilb}
     

}_{X/S}(T) = \left\{ \begin{matrix} Z & \hookrightarrow & X \times_S T & \to & X \\ \downarrow & & \downarrow & & \downarrow \\ T & = & T & \to & S \end{matrix}

Z \to T \text{ is flat}

\right\} / \sim

where the equivalence relation is given by the isomorphism classes of $Z$. This construction is functorial by taking pullbacks of families. Given $f:\; T\text{'}\; \backslash to\; T$, there is a family $f^*Z\; =\; Z\backslash times\_TT\text{'}$ over $T\text{'}$.

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Representability for projective maps If the structure map $X\; \backslash to\; S$ is projective, then this functor is represented by the Hilbert scheme constructed above. Generalizing this to the case of maps of finite type requires the technology of developed by Artin.

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Relative Hilbert scheme for maps of algebraic spaces In its greatest generality, the Hilbert functor is defined for a finite type map of algebraic spaces $f\backslash colon\; X\; \backslash to\; B$ defined over a scheme $S$. Then, the Hilbert functor is defined as

$\backslash underline\{\backslash text\{Hilb\}\}\_\{X/B\}:(Sch/B)^\{op\}\; \backslash to\; Sets$

sending T to

$\backslash underline\{\backslash text\{Hilb\}\}\_\{X/B\}(T)\; =\; \backslash left\backslash \{\; Z\; \backslash subset\; X\backslash times\_BT\; :$

\begin{align} &Z \to T \text{ is flat, proper,} \\ &\text{and of finite presentation} \end{align} \right\}. This functor is not representable by a scheme, but by an algebraic space. Also, if $S\; =\; \backslash text\{Spec\}(\backslash Z)$, and $X\backslash to\; B$ is a finite type map of schemes, their Hilbert functor is represented by an algebraic space.

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Examples of Hilbert schemes

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Fano schemes of hypersurfaces One of the motivating examples for the investigation of the Hilbert scheme in general was the Fano scheme of a projective scheme. Given a subscheme $X\; \backslash subset\; \backslash mathbb\{P\}^n$ of degree $d$, there is a scheme $F\_k(X)$ in $\backslash mathbb\{G\}(k,\; n)$ parameterizing $H\; \backslash subset\; X\; \backslash subset\; \backslash mathbb\{P\}^n$ where $H$ is a $k$-plane in $\backslash mathbb\{P\}^n$, meaning it is a degree one embedding of $\backslash mathbb\{P\}^k$. For smooth surfaces in $\backslash mathbb\{P\}^3$ of degree $d\; \backslash geq\; 3$, the non-empty Fano schemes $F\_k(X)$ are smooth and zero-dimensional. This is because lines on smooth surfaces have negative self-intersection.

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Hilbert scheme of points

Another common set of examples are the Hilbert schemes of $n$-points of a scheme $X$, typically denoted $X^\{n\}$. For a Riemann surface X, $X^\{n\}=S^nX=X^n/S\_n$. For $\backslash mathbb\{P\}^2$ there is a nice geometric interpretation where the boundary loci $B\; \backslash subset\; H$ describing the intersection of points can be thought of parametrizing points along with their tangent vectors. For example, $(\backslash mathbb\{P\}^2)^\{2\}$ is the blowup $Bl\_\{\backslash Delta\}(\backslash mathbb\{P\}^2\backslash times\backslash mathbb\{P\}^2/S\_2)$ of the diagonal modulo the symmetric action.

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Degree d hypersurfaces

The Hilbert scheme of degree k hypersurfaces in $\backslash mathbb\{P\}^n$ is given by the projectivization $\backslash mathbb\{P\}(\backslash Gamma(\backslash mathcal\{O\}(k)))$. For example, the Hilbert scheme of degree 2 hypersurfaces in $\backslash mathbb\{P\}^1$ is $\backslash mathbb\{P\}^2$ with the universal hypersurface given by

$\backslash text\{Proj\}(kx\_0,x\_1\backslash alpha,\backslash beta,\backslash gamma/(\backslash alpha\; x\_0^2\; +\; \backslash beta\; x\_0x\_1\; +\; \backslash gamma\; x\_1^2))\; \backslash subseteq\; \backslash mathbb\{P\}\_\{x\_0,x\_1\}^1\backslash times\backslash mathbb\{P\}^2\_\{\backslash alpha,\backslash beta,\backslash gamma\}$

where the underlying ring is bigraded.

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Hilbert scheme of curves and moduli of curves

For a fixed genus $g$ algebraic curve $C$, the degree of the tri-tensored dualizing sheaf $\backslash omega\_C^\{\backslash otimes\; 3\}$ is globally generated, meaning its Euler characteristic is determined by the dimension of the global sections, so

$\backslash chi(\backslash omega\_C^\{\backslash otimes\; 3\})\; =\; \backslash dim\; H^0(C,\backslash omega\_X^\{\backslash otimes\; 3\})$.

The dimension of this vector space is $5g-5$, hence the global sections of $\backslash omega\_C^\{\backslash otimes\; 3\}$ determine an embedding into $\backslash mathbb\{P\}^\{5g-6\}$ for every genus $g$ curve. Using the Riemann-Roch formula, the associated Hilbert polynomial can be computed as

$H\_C(t)\; =\; 6(g-1)t\; +\; (1-g)$.

Then, the Hilbert scheme

$\backslash text\{Hilb\}\_\{\backslash mathbb\{P\}^\{5g-6\}\}^\{H\_C(t)\}$

parameterizes all genus g curves. Constructing this scheme is the first step in the construction of the moduli stack of algebraic curves. The other main technical tool are GIT quotients, since this moduli space is constructed as the quotient

$\backslash mathcal\{M\}\_g\; =\; U\_g/GL\_\{5g-6\}$,

where $U\_g$ is the sublocus of smooth curves in the Hilbert scheme.

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Hilbert scheme of points on a manifold

"Hilbert scheme" sometimes refers to the punctual Hilbert scheme of 0-dimensional subschemes on a scheme. Informally this can be thought of as something like finite collections of points on a scheme, though this picture can be very misleading when several points coincide.

There is a Hilbert–Chow morphism from the reduced Hilbert scheme of points to the Chow variety of cycles taking any 0-dimensional scheme to its associated 0-cycle. .

The Hilbert scheme $M^\{n\}$ of points on is equipped with a natural morphism to an -th symmetric product of . This morphism is birational for of dimension at most 2. For of dimension at least 3 the morphism is not birational for large : the Hilbert scheme is in general reducible and has components of dimension much larger than that of the symmetric product.

The Hilbert scheme of points on a curve (a dimension-1 complex manifold) is isomorphic to a symmetric power of . It is smooth.

The Hilbert scheme of points on a Complex surface is also smooth (Grothendieck). If $n=2$, it is obtained from $M\backslash times\; M$ by blowing up the diagonal and then dividing by the $\backslash Z/2\backslash Z$ action induced by $(x,y)\; \backslash mapsto\; (y,x)$. This was used by Mark Haiman in his proof of the positivity of the coefficients of some Macdonald polynomials.

The Hilbert scheme of a smooth manifold of dimension 3 or more is usually not smooth.

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Hilbert schemes and hyperkähler geometry

Let be a complex Kähler surface with $c\_1=\; 0$ (K3 surface or a torus). The canonical bundle of is trivial, as follows from the Kodaira classification of surfaces. Hence admits a holomorphic symplectic form. It was observed by Akira Fujiki (for $n=2$) and Arnaud Beauville that $M^\{n\}$ is also holomorphically symplectic. This is not very difficult to see, e.g., for $n=2$. Indeed, $M^\{2\}$ is a blow-up of a symmetric square of . Singularities of $\backslash operatorname\{Sym\}^2\; M$ are locally isomorphic to $\backslash Complex^2\; \backslash times\; \backslash Complex^2/\backslash \{\backslash pm\; 1\backslash \}$. The blow-up of $\backslash Complex^2/\backslash \{\backslash pm\; 1\backslash \}$ is $T^\{*\}\backslash mathbb\{P\}^\{1\}(\backslash Complex)$, and this space is symplectic. This is used to show that the symplectic form is naturally extended to the smooth part of the exceptional divisors of $M^\{n\}$. It is extended to the rest of $M^\{n\}$ by Hartogs' principle.

A holomorphically symplectic, Kähler manifold is hyperkähler, as follows from the Calabi–Yau theorem. Hilbert schemes of points on the K3 surface and on a 4-dimensional torus give two series of examples of hyperkähler manifolds: a Hilbert scheme of points on K3 and a generalized Kummer surface.

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See also

• Quot scheme
• Castelnuovo–Mumford regularity
• Matsusaka's big theorem
• Moduli of algebraic curves
• Moduli space
• Hilbert modular surface
• Siegel modular variety

• Reprinted in

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Examples and applications

• The Number of Twisted Cubics on a Quintic Threefold
• Rational curves on Calabi–Yau threefolds: Verifying mirror symmetry predictions

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External links

Categories: Scheme Theory, Algebraic Geometry, Differential Geometry, Moduli Theory

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