Hilbert scheme

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

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

${\displaystyle 0\to I_{X}\to {\mathcal {O}}_{\mathbb {P} ^{n}}\to {\mathcal {O}}_{X}\to 0}$

we have ${\displaystyle I_{X}^{m}=\Gamma (I_{X}\otimes {\mathcal {O}}_{\mathbb {P} ^{n}}(m))}$ has dimension ${\displaystyle Q(m)-P_{X}(m)}$, where ${\displaystyle Q}$ is the Hilbert polynomial of projective space. This can be shown by tensoring the exact sequence above by the locally flat sheaves ${\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(m)}$, giving an exact sequence where the latter two terms have trivial cohomology, implying the triviality of the higher cohomology of ${\displaystyle I_{X}(m)}$. 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 ${\displaystyle m}$. The ${\displaystyle (Q(m)-P_{X}(m))}$-dimensional space ${\displaystyle I_{X}^{m}}$ is a subspace of the ${\displaystyle Q(m)}$-dimensional space ${\displaystyle S^{n}}$, so represents a point of the Grassmannian ${\displaystyle {\textbf {Gr}}(Q(m)-P_{X}(m),Q(m))}$. This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial ${\displaystyle P_{X}}$ 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 I_X(m) ⊗ S(k) → S(k + m) has rank at most dim(I_X(k + m)) for all positive k, which is equivalent to the vanishing of various determinants. (A more careful analysis shows that it is enough just to take k = 1.)

Given a closed subscheme ${\displaystyle Y\subset \mathbb {P} _{k}^{n}=X}$ over a field with Hilbert polynomial ${\displaystyle P}$, the Hilbert scheme H=Hilb(n, P) has a universal subscheme ${\displaystyle W\subset X\times H}$ flat over ${\displaystyle H}$ such that

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

The tangent space of the point ${\displaystyle [Y]\in H}$ is given by the global sections of the normal bundle ${\displaystyle N_{Y/X}}$; that is,

${\displaystyle T_{[Y]}H=H^{0}(Y,N_{Y/X})}$

For local complete intersections ${\displaystyle Y}$ such that ${\displaystyle H^{1}(Y,N_{X/Y})=0}$, the point ${\displaystyle [Y]\in H}$ is smooth. This implies every deformation of ${\displaystyle Y}$ in ${\displaystyle X}$ is unobstructed.

In the case ${\displaystyle H^{1}(Y,N_{X/Y})\neq 0}$, the dimension of ${\displaystyle H}$ at ${\displaystyle [Y]}$ is greater than or equal to ${\displaystyle h^{0}(Y,N_{X/Y})-h^{1}(Y,N_{X/Y})}$.

In addition to these properties, Francis Sowerby Macaulay (1927) determined for which polynomials the Hilbert scheme ${\displaystyle \mathbf {Hilb} (n,P)}$ is non-empty, and Robin Hartshorne (1966) showed that if ${\displaystyle \mathbf {Hilb} (n,P)}$ 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 d points (more precisely dimension 0, length d subschemes) of a scheme of dimension n to have dimension dn, but if n ≥ 3 its irreducible components can have much larger dimension.

If the structure map ${\displaystyle X\to S}$ 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 algebraic spaces developed by Artin.[2]

In its greatest generality, the Hilbert functor is defined for a finite type map of algebraic spaces ${\displaystyle f\colon X\to B}$ defined over a scheme ${\displaystyle S}$. Then, the Hilbert functor is defined as[3]

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

sending T to

${\displaystyle {\underline {\text{Hilb}}}_{X/B}(T)=\left\{Z\subset X\times _{B}T:{\begin{aligned}&Z\to T{\text{ is flat, proper,}}\\&{\text{and of finite presentation}}\end{aligned}}\right\}}$.

This functor is not representable by a scheme, but by an algebraic space. Also, if ${\displaystyle S={\text{Spec}}(\mathbb {Z} )}$, and ${\displaystyle X\to B}$ is a finite type map of schemes, their Hilbert functor is represented by an algebraic space.

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 ${\displaystyle X\subset \mathbb {P} ^{n}}$ of degree ${\displaystyle d}$, there is a scheme ${\displaystyle F_{k}(X)}$ in ${\displaystyle \mathbb {G} (k,n)}$ parameterizing ${\displaystyle H\subset X\subset \mathbb {P} ^{n}}$ where ${\displaystyle H}$ is a ${\displaystyle k}$-plane in ${\displaystyle \mathbb {P} ^{n}}$, meaning it is a degree one embedding of ${\displaystyle \mathbb {P} ^{k}}$.[4] For smooth surfaces in ${\displaystyle \mathbb {P} ^{3}}$ of degree ${\displaystyle d\geq 3}$, the non-empty Fano schemes ${\displaystyle F_{k}(X)}$ are smooth and zero-dimensional. This is because lines on smooth surfaces have negative self-intersection.[4]

Another common set of examples are the Hilbert schemes of ${\displaystyle n}$-points of a scheme ${\displaystyle X}$, typically denoted ${\displaystyle X^{[n]}}$. For ${\displaystyle \mathbb {P} ^{2}}$ there is a nice geometric interpretation where the boundary loci ${\displaystyle B\subset H}$ describing the intersection of points can be thought of parametrizing points along with their tangent vectors. For example, ${\displaystyle (\mathbb {P} ^{2})^{[2]}}$ is the blowup ${\displaystyle Bl_{\Delta }(\mathbb {P} ^{2}\times \mathbb {P} ^{2}/S_{2})}$ of the diagonal[5] modulo the symmetric action.

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

${\displaystyle {\text{Proj}}(k[x_{0},x_{1}][\alpha ,\beta ,\gamma ]/(\alpha x_{0}^{2}+\beta x_{0}x_{1}+\gamma x_{1}^{2}))\subseteq \mathbb {P} _{x_{0},x_{1}}^{1}\times \mathbb {P} _{\alpha ,\beta ,\gamma }^{2}}$

where the underlying ring is bigraded.

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

${\displaystyle \chi (\omega _{C}^{\otimes 3})=\dim H^{0}(C,\omega _{X}^{\otimes 3})}$.

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

${\displaystyle H_{C}(t)=6(g-1)t+(1-g)}$.

Then, the Hilbert scheme

${\displaystyle {\text{Hilb}}_{\mathbb {P} ^{5g-6}}^{H_{C}(t)}}$

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

${\displaystyle {\mathcal {M}}_{g}=[U_{g}/GL_{5g-6}]}$,

where ${\displaystyle U_{g}}$ is the sublocus of smooth curves in the Hilbert scheme.

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