Hilbert scheme (${\mathrm{Hilb}}^P(X)$)

For a projective scheme $X$ and a fixed Hilbert polynomial $P$, the Hilbert scheme parametrizes flat families of closed subschemes of $X$ with that polynomial. This turns a family of embedded algebraic curves into an algebraic parameter space.

For a projective scheme $X$ over a field $k$, the Hilbert functor assigns to a $k$-scheme $T$ the set of closed subschemes $W\subseteq X{\times }_{k}T$ which are of finite presentation, proper and flat over $T$. Pullback of families makes it a contravariant functor. After choosing a very ample line bundle, requiring a fixed fibre Hilbert polynomial $P$ gives the subfunctor represented by ${\mathrm{Hilb}}^P(X)$. The unrestricted functor is represented by the disjoint union of these Hilbert schemes.

A first-order embedded deformation of a closed subscheme $Z\subseteq X$ is a closed subscheme $Z_{\epsilon }\subseteq X{\times }_k\mathrm{Spec}(k[\epsilon ]/({\epsilon }^2))$ flat over the dual numbers and with special fibre equal to the given embedded $Z$. It is a tangent vector of the Hilbert scheme, not a deformation modulo arbitrary automorphisms of the ambient scheme. These deformations are naturally the global sections of the normal sheaf.

For a closed subscheme $Z\subseteq X$ with ideal sheaf $\mathcal{I}$, one has $T_{[Z]}\mathrm{Hilb}(X)\cong {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{I},i_{\ast }{\mathcal{O}}_Z)\cong H^0(Z,{\mathcal{N}}_{Z/X})$. Locally a flat deformation ideal $J\subseteq A[\epsilon ]/({\epsilon }^2)$ satisfies $J\cap \epsilon A=\epsilon I$. Lifting $f\in I$ to $f+\epsilon g\in J$ gives a well-defined map $f\mapsto g\mathrm{mod}I$ killing $I^2$. Conversely the graph condition $g\mathrm{mod}I=\varphi (f)$ defines an ideal with flat quotient by the flatness criterion over dual numbers. These operations commute with restriction and are inverse, giving the global identification without a smoothness assumption.

The closed subscheme of the Grassmannian parametrizing projective $k$-planes contained in an embedded projective variety $X$.

The Fano scheme for projective lines. For $X\subseteq {\mathbb{P}}^3$, it is a closed subscheme of $\mathrm{Gr}(2,4)$ and hence is projective. Requiring the defining homogeneous polynomials of $X$ to vanish along a line gives its closed defining conditions.

Cubic forms in four variables form ${\mathbb{P}}^{19}$. Lines form the four-dimensional Grassmannian $G=\mathrm{Gr}(2,4)$. The incidence pairs $(\ell ,[F])$ with $F{|}_{\ell }=0$ form a projective-space bundle with fibre ${\mathbb{P}}^{15}$ over $G$, since restriction to a line imposes four independent linear conditions. Hence the incidence variety is irreducible of dimension 19. For $F=x_0^2{x}_2+x_1^2{x}_3$, a nearby line written $x_2=a_0{x}_0+a_1{x}_1$, $x_3=b_0{x}_0+b_1{x}_1$ lies on $F$ precisely when all four parameters vanish. This is an isolated point of that fibre. The fiber dimension theorem forces the image to have dimension 19, and projectivity makes the image closed. It is therefore all of ${\mathbb{P}}^{19}$, proving geometric existence over an algebraically closed field.

A projective bundle with fibre ${\mathbb{P}}^{15}$ over the four-dimensional Grassmannian $\mathrm{Gr}(2,4)$. The condition of containing a fixed line imposes four independent linear conditions on the twenty cubic coefficients. The projection to ${\mathbb{P}}^{19}$ is proper. The line $\{x_2=x_3=0\}$ is an isolated point over $x_0^2{x}_2+x_1^2{x}_3$; isolated fibre point forces dominance in equal dimensions then proves that every cubic surface contains a line.

For a line $L$ on a degree-$d$ smooth surface in ${\mathbb{P}}^3$, the adjunction formula gives $L^2=2-d$. If $d\ge 3$, all lines are negative curves, hence there are only countably many by countability of negative curves. Their projective Fano scheme of lines cannot have a positive-dimensional complex component, since such a component has uncountably many points. A zero-dimensional projective scheme of finite type has finitely many points, proving the assertion.