Embedded flat deformations of a smooth hypersurface are deformations of its equation

Let $F\in (S{\otimes }_k{A}^{\mathrm{\prime }}{)}_d$ with $F-f\in \mathfrak{m}(S{\otimes }_k{A}^{\mathrm{\prime }}{)}_d$, where $\mathfrak{m}={\mathfrak{m}}_{A^{\mathrm{\prime }}}$. On the chart $D_{+}(x_i)$ put $B_i=(S{\otimes }_k{A}^{\mathrm{\prime }}{)}_{(x_i)}$ and $F_i=F\mathrm{/}{x}_i^d\in B_i$; then $B_i\mathrm{/}\mathfrak{m}{B}_i=S_{(x_i)}$ is a domain, and the reduction ${\bar{F}}_i=f\mathrm{/}{x}_i^d$ is nonzero because $f\mathrm{\ne }0$ and $S$ is a domain, hence is a nonzerodivisor on $B_i\mathrm{/}\mathfrak{m}{B}_i$. Since ${\mathfrak{m}}^N=0$ for some $N$ by [F9], $F_i$ is a nonzerodivisor on $B_i$: if $F_{i}h=0$ with $h\mathrm{\ne }0$, choose $t$ maximal with $h\in {\mathfrak{m}}^t{B}_i$ and read the equation in ${\mathfrak{m}}^t{B}_i\mathrm{/}{\mathfrak{m}}^{t+1}{B}_i\cong ({\mathfrak{m}}^t\mathrm{/}{\mathfrak{m}}^{t+1}){\otimes }_k(B_i\mathrm{/}\mathfrak{m}{B}_i)$, an identification valid because $B_i$ is flat over $A^{\mathrm{\prime }}$; there $F_i$ acts as the nonzerodivisor ${\bar{F}}_i$, the class of $h$ is nonzero, and the product vanishes, a contradiction.

Now let $Y\subseteq {\mathbb{P}}_{A^{\mathrm{\prime }}}^n$ be closed and flat over $A^{\mathrm{\prime }}$ with reduction the embedded equation family $Z(F_A)$, where $F_A$ reduces to $f$ on the special fibre. On the chart $D_{+}(x_i)$ let $J_i\subseteq B_i$ be the ideal of $Y\cap D_{+}(x_i)$; then $B_i\mathrm{/}{J}_i$ is flat over $A^{\mathrm{\prime }}$ by [F3], the ring $B_i$ is Noetherian by [F9] so $J_i$ is finitely generated as a $B_i$-module; no finite-generation assertion over the Artin base is needed. The base change to $A$ is the affine chart of $Z(F_A)$, so $(J_i+I{B}_i)\mathrm{/}I{B}_i=(F_A\mathrm{/}{x}_i^d)$ inside $B_i\mathrm{/}I{B}_i$; choose $g_i\in J_i$ whose reduction modulo $I{B}_i$ is $F_A\mathrm{/}{x}_i^d$. Then $J_i+I{B}_i=(g_i)+I{B}_i$.

With the notation of step 1.1, the quotient $B_i\mathrm{/}{F}_i{B}_i$ is flat over $A^{\mathrm{\prime }}$: for an ideal $\mathfrak{a}\subseteq A^{\mathrm{\prime }}$ the kernel of $\mathfrak{a}{\otimes }_{A^{\mathrm{\prime }}}(B_i\mathrm{/}{F}_i{B}_i)\to B_i\mathrm{/}{F}_i{B}_i$ is $(\mathfrak{a}{B}_i\cap F_i{B}_i)\mathrm{/}\mathfrak{a}{F}_i{B}_i$, so by [F4] it suffices to show $\mathfrak{a}{B}_i\cap F_i{B}_i=\mathfrak{a}{F}_i{B}_i$. If $x=F_{i}y$ with $x\in \mathfrak{a}{B}_i$ and $y\mathrm{\notin }\mathfrak{a}{B}_i$, then the class of $y$ in $B_i\mathrm{/}\mathfrak{a}{B}_i$ is nonzero, and choosing $t$ maximal with $y\in {\mathfrak{m}}^t{B}_i+\mathfrak{a}{B}_i$ gives a nonzero class in ${\mathfrak{m}}^t(B_i\mathrm{/}\mathfrak{a}{B}_i)\mathrm{/}{\mathfrak{m}}^{t+1}(B_i\mathrm{/}\mathfrak{a}{B}_i)\cong (({\mathfrak{m}}^t+\mathfrak{a})\mathrm{/}({\mathfrak{m}}^{t+1}+\mathfrak{a})){\otimes }_{k}k[x_{\mathrm{\ell }}^{(i)}]$ killed by the nonzerodivisor ${\bar{F}}_i$, a contradiction. Hence $B_i\mathrm{/}{F}_i{B}_i$ is flat over $A^{\mathrm{\prime }}$, and the chart subscheme $Z(F)\cap D_{+}(x_i)=\mathrm{Spec}(B_i\mathrm{/}{F}_i{B}_i)$ is flat over $A^{\mathrm{\prime }}$; gluing over the charts, $Z(F)$ is flat over $A^{\mathrm{\prime }}$ by [F3]. Base change reduces the chart equation to $F_A\mathrm{/}{x}_i^d$, so the intermediate fibre is $Z(F_A)$; when $F\equiv f$ modulo $I$ this is exactly the trivial embedded $X{\times }_k\mathrm{Spec}A$. Matching $f$ only modulo the maximal ideal ensures the special fibre is $X$, without forcing triviality over $A$. This proves claim (1).

With $g_i$ as in step 1.2, we have $J_i=(g_i)$. Indeed, multiplication $I{\otimes }_{A^{\mathrm{\prime }}}(B_i\mathrm{/}{J}_i)\to B_i\mathrm{/}{J}_i$ is injective by flatness and [F4]. For $j\in J_i\cap I{B}_i$, choose a tensor in $I{\otimes }_{A^{\mathrm{\prime }}}{B}_i$ whose product is $j$. Its image in $I{\otimes }_{A^{\mathrm{\prime }}}(B_i\mathrm{/}{J}_i)$ is zero by that injectivity; right exactness of tensor therefore makes it the image of a tensor in $I{\otimes }_{A^{\mathrm{\prime }}}{J}_i$. Its product lies in $I{J}_i$, proving $J_i\cap I{B}_i=I{J}_i$. Hence every $j\in J_i$ lies in $(g_i)+I{J}_i$, so $J_i\mathrm{/}(g_i)=I\cdot J_i\mathrm{/}(g_i)$; the module $J_i\mathrm{/}(g_i)$ is finitely generated over $B_i$, and $I{B}_i$ is nilpotent, hence lies in the Jacobson radical of $B_i$, so Nakayama [F5] applied over $B_i$ gives $J_i=(g_i)$. The element $g_i$ is a nonzerodivisor: $g_i$ reduces to $f\mathrm{/}{x}_i^d\mathrm{\ne }0$ modulo $\mathfrak{m}{B}_i$, so the filtration argument of step 1.1 applies verbatim. Thus $Y$ is a relative effective Cartier divisor on every chart, hence globally, with local equations $g_i$; the degree is $d$ because the special fibre is $X$, so [F8] applies.

Let $B_{ij}=B_i[(x_j\mathrm{/}{x}_i{)}^{-1}]$ be the overlap ring; the hypotheses of step 2.2 give $g_i\mathrm{/}{g}_j\in B_{ij}^{\times }$ and $g_i\equiv F_A\mathrm{/}{x}_i^d$, $g_j\equiv F_A\mathrm{/}{x}_j^d$ modulo $I{B}_{ij}$. Define the unit $c_{ij}=u_{ij}(x_i\mathrm{/}{x}_j{)}^d$ with $u_{ij}=g_i\mathrm{/}{g}_j$. Then $c_{ij}\equiv (x_j\mathrm{/}{x}_i{)}^d(x_i\mathrm{/}{x}_j{)}^d=1$ modulo $I{B}_{ij}$, and $c_{ij}{c}_{jk}=c_{ik}$ because $u_{ij}{u}_{jk}=u_{ik}$; so $c=(c_{ij})$ is a multiplicative Cech $1$-cocycle of the standard cover with values in $1+I\mathcal{O}$, and $1+a\mapsto a$ identifies this group sheaf with $I{\mathcal{O}}_{{\mathbb{P}}_{A^{\mathrm{\prime }}}^n}$, since $I^2=0$. The latter sheaf, on the common underlying space, is $I{\otimes }_k{\mathcal{O}}_{{\mathbb{P}}_k^n}$: flatness of the polynomial chart rings identifies $I{B}_i$ with $I{\otimes }_{A^{\mathrm{\prime }}}{B}_i=I{\otimes }_{k}k[x_{\mathrm{\ell }}^{(i)}]$, compatibly on overlaps. It is a finite direct sum of copies of ${\mathcal{O}}_{{\mathbb{P}}_k^n}$, whose Cech $H^1$ vanishes by [F2] and [F6]; hence there are ${\lambda }_i\in 1+I{B}_i$ with $c_{ij}={\lambda }_i\mathrm{/}{\lambda }_j$.

Define $F_i=g_i\mathrm{/}{\lambda }_i\in B_i$. Then $F_i\equiv g_i\equiv F_A\mathrm{/}{x}_i^d$ modulo $I{B}_i$, and on overlaps $F_i\mathrm{/}{F}_j=(g_i\mathrm{/}{g}_j)({\lambda }_j\mathrm{/}{\lambda }_i)=u_{ij}\mathrm{/}{c}_{ij}=(x_j\mathrm{/}{x}_i{)}^d$, so $x_i^d{F}_i=x_j^d{F}_j$. By [F1] the family $(F_i)$ glues to a global section $F\in H^0({\mathbb{P}}_{A^{\mathrm{\prime }}}^n,\mathcal{O}(d))=(S{\otimes }_k{A}^{\mathrm{\prime }}{)}_d$, whose chart components are $F_i$; since $F_i\equiv F_A\mathrm{/}{x}_i^d$ modulo $I$, the global sections $F$ and $F_A$ agree modulo $I$, and on each chart the ideals $(F_i)=(g_i)$ coincide because ${\lambda }_i$ is a unit, so $Z(F)=Y$ globally. This proves the existence in claim (2).

If $Z(F)=Z(F^{\mathrm{\prime }})=Y$ with $F,F^{\mathrm{\prime }}$ both lifts of $f$ as above, then on each chart $F_i^{\mathrm{\prime }}={\mu }_i{F}_i$ for a unit ${\mu }_i\in B_i^{\times }$, and ${\mu }_i={\mu }_j$ on overlaps because both sides multiply the nonzerodivisor $F_i$ to give $F_i^{\mathrm{\prime }}$; hence the ${\mu }_i$ glue to a global unit $\mu \in \mathrm{\Gamma }({\mathbb{P}}_{A^{\mathrm{\prime }}}^n,{\mathcal{O}}_{A^{\mathrm{\prime }}}^{\times })$, and $F^{\mathrm{\prime }}=\mu F$ with $\mu \in A^{\mathrm{\prime }}\subseteq B_i$ (as $H^0(\mathcal{O})=A^{\mathrm{\prime }}$ by [F2]). Since $\mu$ is a unit in every $B_i$, in particular $\mu$ reduces to a nonzero constant on the special fibre, so $\mu$ is a unit of the local ring $A^{\mathrm{\prime }}$ by [F7]. This completes the uniqueness in claim (2).

Claims (1) and (2) establish the equation description across every small extension, including arbitrary nontrivial reductions $F_A$. Every local Artin $R$ with residue field $k$ admits a finite tower of such extensions: choose a one-dimensional subspace of the last nonzero maximal-ideal power, which is an ideal annihilated by that maximal ideal, quotient by it, and repeat until $k$. Inducting along this finite tower starts with $f$ over $k$ and applies claim (2) to the chosen equation of each preceding reduction. Thus for every such $R$, the claims establish a canonical bijection between isomorphism classes of flat closed subschemes $Y\subseteq {\mathbb{P}}_R^n$ with special fibre $X$ and classes of lifts $F\in (S{\otimes }_{k}R{)}_d$ modulo $R^{\times }$; the construction is natural in $R$ because both steps 4.1 and 5.1 are computed from the chart data and commute with base change $R\to R^{\mathrm{\prime }}$. Hence the embedded deformation functor of $X$ in ${\mathbb{P}}^n$ is canonically isomorphic to $R\mapsto \{F\in (S{\otimes }_{k}R{)}_d:F\equiv f<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>{\mathfrak{m}}_R\}\mathrm{/}(R^{\times })$. By definition of the formal completion of $\mathbb{P}(S_d^{\vee })$ at the point $[f]$, an $R$-point of that completion is exactly a morphism $\mathrm{Spec}R\to \mathbb{P}(S_d^{\vee })$ lifting $[f]$, which (as $R$ is local, so rank-one quotients are free) is a rank-one direct summand of $S_d{\otimes }_{k}R$, equivalently a lift of $f$ modulo $R^{\times }$ (the quotient convention of Projective bundle in the quotient convention uses the dual vector space); this identifies the functor with the functor of points of the formal completion. Explicitly, choose a basis $f,e_1,\dots ,e_N$ of $S_d$. Each class has a unique representative $f+{\sum }_i{a}_i{e}_i$ with $a_i\in {\mathfrak{m}}_R$. Thus the representing complete local ring is $k[\text{\hspace{0.17em}⁣}[t_1,\dots ,t_N]\text{\hspace{0.17em}⁣}]$: continuous local maps to $R$ send $t_i$ to $a_i$, and power series evaluate by finite sums because ${\mathfrak{m}}_R$ is nilpotent.

The functor of step 6.1 is formally smooth: given a small extension $A^{\mathrm{\prime }}\to A$ and a lift $F$ of $f$ over $A$, choose any lift $\tilde{F}$ of $F$ to $(S{\otimes }_k{A}^{\mathrm{\prime }}{)}_d$; then $\tilde{F}\equiv f$ modulo ${\mathfrak{m}}_{A^{\mathrm{\prime }}}$ automatically because $\tilde{F}-F\in I(S{\otimes }_k{A}^{\mathrm{\prime }})\subseteq {\mathfrak{m}}_{A^{\mathrm{\prime }}}(S{\otimes }_k{A}^{\mathrm{\prime }})$ and $F\equiv f$ modulo ${\mathfrak{m}}_A(S{\otimes }_{k}A)$. The same computation applies to classes modulo units: if $G$ is any lift of the class $[F]$ and $u\in A^{\times }$ with $G\equiv uF$ modulo $I$, lift $u$ to a unit $\tilde{u}\in (A^{\mathrm{\prime }}{)}^{\times }$ and replace $G$ by ${\tilde{u}}^{-1}G$. Hence restriction is surjective on every small extension, and the functor is unobstructed.

For the dual numbers $R=k[ϵ]$ the lifts are $F=f+ϵg$ with $g\in S_d$, and multiplication by a unit of $k[ϵ]$ followed by renormalisation of the $ϵ$-free part to $f$ (division by the unit $\lambda$, $\lambda \mathrm{\ne }0$) replaces $g$ by $g+\mu f$; hence classes correspond bijectively to $S_d\mathrm{/}k\cdot f\cong (S\mathrm{/}(f){)}_d$, which is $H^0(X,{\mathcal{O}}_X(d))$ by Cohomology of twists on a smooth hypersurface(2) and has dimension $\left(\genfrac{}{}{0px}{}{n+d}{n}\right)-1$. By steps 6.1 and 7.1 every class extends to every small extension, so this is the tangent space at the trivial deformation.

Steps 1.1 and 2.1 prove claim (1); steps 1.2, 2.2, 3.1, 4.1 and 5.1 prove claim (2); steps 6.1 and 7.1 prove claim (3); and step 8.1 proves claim (4). The Axiom of Choice is inherited from the cohomology, Cech-comparison, Nakayama and Artinian-ring suppliers.