First-order deformations of a plane conic and of a quadric surface

Flatness and completeness of the equation description. By [F4] every $g\in S_2$ gives a flat $k[ϵ]$-subscheme $Z(f+ϵg)\subseteq {\mathbb{P}}_{k[ϵ]}^2$ with special fibre $C$, and by [F1] every first-order embedded deformation of $C$ is of this form. This is claim (1).

Every fibre with Hilbert polynomial $2t+1$ is a conic. Over a field $K$, let $J\subset K[x_0,x_1,x_2]$ be the homogeneous ideal of such a subscheme $Y$. It is nonzero, since otherwise $Y={\mathbf{P}}_K^2$ would have a quadratic Hilbert polynomial. Choose finite homogeneous generators by [F6], and let $h$ be their greatest common divisor, homogeneous of degree $e$. At every height-one prime the localization of $J$ equals $(h)$, because the exponent in the gcd is the minimum exponent in the generators. Thus $\mathcal{Q}=\widetilde{(h)}\mathrm{/}\tilde{J}$ is supported on finitely many points of ${\mathbf{P}}_K^2$: after dividing the generators by $h$ their ideal lies in no height-one prime by [F6]. The exact sequence $0\to \mathcal{Q}\to {\mathcal{O}}_Y\to {\mathcal{O}}_{Z(h)}\to 0$ gives $P_Y(t)=et+e(3-e)\mathrm{/}2+\mathrm{\ell }$, where the divisor polynomial follows from $0\to \mathcal{O}(-e)\to \mathcal{O}\to {\mathcal{O}}_{Z(h)}\to 0$ and [F3], and $\mathrm{\ell }={\dim }_K{H}^0(\mathcal{Q})$ is nonnegative. Indeed a coherent sheaf with finite support is a finite module on a finite Artinian scheme, its twists are locally isomorphic to itself, and its positive cohomology vanishes; hence its polynomial is that nonnegative dimension. The coefficient of $t$ forces $e=2$, and then the constant term forces $\mathrm{\ell }=0$. Therefore $\mathcal{Q}=0$ and $Y=Z(h)$ scheme theoretically. This includes double lines, reducible conics and arbitrary residue fields.

The unit orbit and the coefficient relation. Let $F=f+ϵg$ and $F^{\mathrm{\prime }}=f+ϵg^{\mathrm{\prime }}$ be two lifts of $f$. By [F4] they define the same deformation exactly when $F^{\mathrm{\prime }}=uF$ for a unit $u=\lambda +ϵ\mu \in k[ϵ{]}^{\times }$, $\lambda \mathrm{\ne }0$. Comparing $ϵ$-free parts gives $1=\lambda$, so $\lambda =1$ and comparing $ϵ$-parts gives $g^{\mathrm{\prime }}=g+\mu f$, i.e. $g^{\mathrm{\prime }}-g\in k\cdot f$; conversely if $g^{\mathrm{\prime }}=g+\mu f$ then $F^{\mathrm{\prime }}=(1+ϵ\mu )F$ with $1+ϵ\mu$ a unit (inverse $1-ϵ\mu$), so the deformations agree. Hence the classes of first-order embedded deformations are in bijection with $S_2\mathrm{/}k\cdot f$, i.e. with $(S\mathrm{/}(f){)}_2$, the trivial deformation corresponding to $g=0$; the dimension is ${\dim }_k{S}_2-1=5$ by [F3], and by [F2] this equals the dimension of $H^0(C,{\mathcal{N}}_{C\mathrm{/}{\mathbb{P}}^2})\cong H^0(C,{\mathcal{O}}_C(2))$. This proves claim (2).

Recover the universal equation line. Let $\pi :{\mathbf{P}}_H^2\to H$ and let $\mathcal{I}$ be the ideal of the universal family of [F5]. All fibres are conics by step 1.2, so [F8] makes this family a relative effective Cartier divisor; consequently $\mathcal{I}$ is invertible with fibre $\mathcal{O}(-2)$. Set $\mathcal{L}=\mathcal{I}(2)$. It is coherent and flat over $H$, and every fibre is ${\mathcal{O}}_{{\mathbf{P}}^2}$, with $h^0=1$ and $h^1=0$ by [F3]. Apply [F7] in degree one: the fibre map is surjective since its target is zero, so $R^1{\pi }_{\ast }\mathcal{L}$ commutes with base change and has zero fibres, hence is zero by Nakayama. It is therefore locally free of rank zero, and the degree-one locally free criterion in [F7] gives surjectivity of the degree-zero fibre map. The degree-zero conclusions of [F7] now show that $E={\pi }_{\ast }\mathcal{L}$ is an invertible sheaf, commutes with arbitrary base change, and has evaluation ${\pi }^{\ast }E\to \mathcal{L}$ an isomorphism: on every fibre this is the evaluation of the one-dimensional space of sections of $\mathcal{O}$, and a map of line bundles that is nonzero at every point is invertible. Pushing the inclusion $\mathcal{L}\subset \mathcal{O}(2)$ down gives $E\hookrightarrow S_2{\otimes }_k{\mathcal{O}}_H$. This is a line subbundle: at each point a coefficient of the fibre equation is nonzero, and locally that coefficient is a unit and splits the inclusion. By the rank-one quotient convention of Projective bundle in the quotient convention, dualizing this line subbundle gives a rank-one quotient of $S_2^{\vee }{\otimes }_k{\mathcal{O}}_H$, hence a morphism $q:H\to \mathbf{P}(S_2^{\vee })$.

Unobstructedness. By [F2] and [F1], $H^1(C,{\mathcal{O}}_C(2))=0$ and the equation functor is formally smooth and unobstructed, so every first-order embedded deformation of $C$ extends over a small extension. We now prove the separate global Hilbert-scheme assertion, following the inverse coefficient-family constructions in Nitsure Section 1, Examples (4).

Construct the inverse family. The tautological coefficient line on $P=\mathbf{P}(S_2^{\vee })$ defines a map ${\mathcal{O}}_P(-1)⊠{\mathcal{O}}_{{\mathbf{P}}^2}(-2)\to {\mathcal{O}}_{{\mathbf{P}}_P^2}$ and its zero scheme ${\mathcal{Z}}_P$. This family is locally principal, and at every fibre its equation is a nonzero quadratic, hence a nonzerodivisor in the polynomial ring over that residue field. The locally principal clause of [F8] therefore makes it a relative effective Cartier divisor, in particular flat over $P$. It is finitely presented, and every fibre has Hilbert polynomial $2t+1$ by the divisor calculation in step 1.2. The representing property [F5] supplies a morphism $u:P\to H$. Recovering the equation line of this tautological family by step 2.2 gives exactly ${\mathcal{O}}_P(-1)\subset S_2\otimes {\mathcal{O}}_P$: its ideal twisted by $2$ is the pullback of that line, and ${\pi }_{\ast }{\mathcal{O}}_{{\mathbf{P}}_P^2}={\mathcal{O}}_P$ by [F3]. Therefore $q\circ u={\mathrm{i}\mathrm{d}}_P$. Conversely, the evaluation isomorphism of step 2.2 identifies the ideal of the universal family with ${\pi }^{\ast }E\otimes \mathcal{O}(-2)$ and its inclusion with the coefficient line defining $q$. Pulling ${\mathcal{Z}}_P$ back by $q$ consequently recovers the universal ideal exactly. The Hilbert representing property gives $u\circ q={\mathrm{i}\mathrm{d}}_H$.

Thus $H\cong \mathbf{P}(S_2^{\vee })={\mathbf{P}}_k^5$ as schemes carrying their universal families. Because [F5] represents the Hilbert functor on all test schemes, the inverse scheme maps in step 3.2 prove the global functor identification on arbitrary $k$-schemes, including non-Noetherian bases. Projective space is smooth of dimension five at $[C]$ by its affine-space charts. This proves the global part of claim (3), without restricting the Hilbert scheme to smooth conics or inferring its scheme structure from a tangent calculation.

The quadric surface and the general formula. For a smooth quadric surface $Q=Z(f)\subseteq {\mathbb{P}}_k^3$ the same computation with $n=3$ gives tangent space $(S\mathrm{/}(f){)}_2$ of dimension ${\dim }_k{S}_2-1=10-1=9$ by [F3], matching ${\dim }_k{H}^0(Q,{\mathcal{O}}_Q(2))=9$ and the classical table $h^0({\mathcal{N}}_Q)=9$ of Hartshorne Chapter 3; for a general smooth hypersurface of degree $d$ in ${\mathbb{P}}_k^n$ the formula is ${\dim }_k(S\mathrm{/}(f){)}_d=\left(\genfrac{}{}{0px}{}{n+d}{n}\right)-1$ by [F1] and [F2]. The Axiom of Choice is inherited from the cohomology, Hilbert-representability, base-change and Nakayama suppliers.