Deformation Theory of Schemes and Obstruction Spaces — Examples

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.

The affine line has vanishing deformation tangent space. The affine line is quasi-compact and separated, so it satisfies the geometric hypotheses of [F4]. By [F1], $L_{X\mathrm{/}k}\simeq {\mathrm{\Omega }}_{X\mathrm{/}k}^1[0]\cong {\mathcal{O}}_X[0]$, so [F2] and [F3] give ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X)\cong H^1({\mathbb{A}}_k^1,{\mathcal{O}}_{{\mathbb{A}}_k^1})=0$.

A non-identity infinitesimal automorphism. Consider the trivial deformation $\mathrm{Spec}k[ϵ][x]$ over the dual numbers and the $k[ϵ]$-algebra endomorphism $\sigma \text{\hspace{0.17em}⁣}:k[ϵ][x]\to k[ϵ][x]$, $x\mapsto x+ϵx^2$. It is a well-defined $k[ϵ]$-algebra map because $x+ϵx^2$ is a polynomial; its inverse is $\tau \text{\hspace{0.17em}⁣}:x\mapsto x-ϵx^2$, since $\sigma (\tau (x))=x+ϵx^2-ϵ(x+ϵx^2{)}^2=x+ϵx^2-ϵx^2=x$ and likewise $\tau (\sigma (x))=x$ because ${ϵ}^2=0$. Moreover $\sigma$ reduces to the identity modulo $ϵ$ but $\sigma (x)-x=ϵx^2\mathrm{\ne }0$, so $\sigma$ is a non-identity automorphism of the trivial deformation reducing to the identity.

All deformation classes are trivial. Since ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(L_{X\mathrm{/}k},{\mathcal{O}}_X)=0$ by step 1.1, [F4] gives that for every local Artin $k$-algebra $A$ with residue field $k$ every deformation of ${\mathbb{A}}_k^1$ over $\mathrm{Spec}A$ is isomorphic to the trivial deformation; this confirms the hypothesis and the ${\mathrm{Ext}}^1$-part of the refuted statement.

Identification with a derivation and conclusion. By [F5] the infinitesimal automorphism group of the trivial deformation is ${\mathrm{Ext}}_{{\mathcal{O}}_X}^0(L_{X\mathrm{/}k},{\mathcal{O}}_X)={\mathrm{Der}}_k(k[x],k[x])=k[x]{\mathrm{\partial }}_x$, which is infinite-dimensional; the automorphism $\sigma$ of step 1.2 corresponds to the derivation $x^2{\mathrm{\partial }}_x$. Hence the deformation groupoid is not trivial and the trivial deformation is not rigid, although its tangent space vanishes by step 1.1 and all deformation classes are trivial by step 2.1. The correct statement is rigidity of isomorphism classes (Vanishing of the deformation tangent space forces rigidity of deformation classes); rigidity of the groupoid would require the additional vanishing of ${\mathrm{Ext}}^0$, which fails here.