A proper nonprojective scheme from glued projective spaces

Write $U_i^{(m)}$ for the standard charts of $X_m$, with coordinates $u_{ij}=x_j\mathrm{/}{x}_i$ ($j\mathrm{\ne }i$), so that $A_i=k[u_{ij}:j\mathrm{\ne }i]$; the charts cover $X_m$ and on $U_i^{(m)}\cap U_j^{(m)}$ one has $u_{i\mathrm{\ell }}=u_{j\mathrm{\ell }}\mathrm{/}{u}_{ji}$ for $\mathrm{\ell }\mathrm{\ne }i,j$ and $u_{ij}=1\mathrm{/}{u}_{ji}$ by [F1]. Consequently, for a homogeneous form $F$ of degree $d$ with dehomogenizations $F_i$ and $F_j$ at $i$ and $j$, one has $F_j=u_{ji}^d{F}_i=u_{ij}^{-d}{F}_i$ on the overlap, a unit multiple of $F_i$; thus the chart ideals generated by the dehomogenizations of a fixed finite list of homogeneous forms have localizations that correspond under the transition isomorphisms, and gluing data built from them are compatible.

Define ${\phi }_m:{\mathbb{P}}_k^1\to X_m$ on the two standard charts of the line as follows: with $w$ the coordinate on $\mathrm{Spec}k[w]=U_0^{({\mathbb{P}}^1)}$ use the ring map $k[u_{30},u_{31},u_{32}]\to k[w]$, $u_{30}\mapsto 0$, $u_{31}\mapsto -w$, $u_{32}\mapsto -w^2$, whose target ideal $(u_{30},u_{31}^2+u_{32})$ is generated by the equations of $C_m\cap U_3^{(m)}$; with $w^{\mathrm{\prime }}$ the coordinate on the other chart use $k[u_{20},u_{21},u_{23}]\to k[w^{\mathrm{\prime }}]$, $u_{20}\mapsto 0$, $u_{21}\mapsto w^{\mathrm{\prime }}$, $u_{23}\mapsto -w^{\mathrm{\prime }2}$, with target ideal $(u_{20},u_{21}^2+u_{23})$. On the overlap $w{w}^{\mathrm{\prime }}=1$ the two composites agree, because the point is $(x_0:x_1:x_2:x_3)=(0:w:w^2:-1)$ in the first chart and $(0:w^{\mathrm{\prime }}:1:-w^{\mathrm{\prime }2})$ in the second, and these are the same projective point when $w{w}^{\mathrm{\prime }}=1$; by [F4] they glue to a $k$-morphism ${\phi }_m:{\mathbb{P}}_k^1\to X_m$ with image in $C_m$. This morphism is an isomorphism onto $C_m$: it maps the two source charts isomorphically onto $C_m\cap U_3^{(m)}$ and $C_m\cap U_2^{(m)}$ with inverses $w=-u_{31}$ and $w^{\mathrm{\prime }}=u_{21}$, and those two pieces cover $C_m$, since a point of $C_m\cap U_1^{(m)}$ with $u_{12}=u_{13}=0$ would satisfy $1+u_{12}{u}_{13}=1\mathrm{\ne }0$.

Let $L_m\subseteq X_m$ be the closed subscheme with chart pieces $L_m\cap U_0^{(m)}=V(u_{02},u_{03})$, $L_m\cap U_1^{(m)}=V(u_{12},u_{13})$ and $L_m\cap U_2^{(m)}=L_m\cap U_3^{(m)}=\mathrm{\varnothing }$. By step 1.1 the localized ideals $(u_{02},u_{03})$ and $(u_{12},u_{13})$ correspond on $U_0^{(m)}\cap U_1^{(m)}$, so the two affine pieces glue along their overlap to a scheme $L_m$ mapping to $X_m$ by a morphism whose restrictions to the chart pieces are closed immersions; [F2] supplies the gluing and [F3] makes the morphism $L_m\to X_m$ a closed immersion. Moreover $L_m\cap U_0^{(m)}=\mathrm{Spec}k[u_{01}]$ and $L_m\cap U_1^{(m)}=\mathrm{Spec}k[u_{10}]$ are glued by $u_{01}=1\mathrm{/}{u}_{10}$, which by [F1] is exactly the standard two-chart presentation of ${\mathbb{P}}_k^1$, so [F2] gives a canonical isomorphism ${\psi }_m:L_m\to {\mathbb{P}}_k^1$ with ${\psi }_m(u_{01})$ the standard coordinate; $\mathrm{\mid }{L}_m\mathrm{\mid }$ is the set of points with $x_2=x_3=0$.

Let $C_m\subseteq X_m$ be the closed subscheme with chart pieces $C_m\cap U_0^{(m)}=\mathrm{\varnothing }$, $C_m\cap U_1^{(m)}=V(u_{10},1+u_{12}{u}_{13})$, $C_m\cap U_2^{(m)}=V(u_{20},u_{21}^2+u_{23})$ and $C_m\cap U_3^{(m)}=V(u_{30},u_{31}^2+u_{32})$; by step 1.1 these are the dehomogenizations of $x_0$ and $x_1^2+x_2{x}_3$, their localizations correspond on every overlap, and [F2] with [F3] makes $C_m$ a closed subscheme of $X_m$ lying in the plane $x_0=0$. On the three charts of that plane the conic has equations $1+bc$, $a^2+c$ and $a^2+b$ for the two remaining ratio coordinates $a,b,c$; the first partials are $(c,b)$, $(2a,1)$ and $(2a,1)$, and a singular point would have to make the equation and both partials vanish: on the first chart $b=c=0$ would force $1=0$, and on the other two charts the second partial is $1$. So $C_m$ is a nonsingular plane conic in the sense of [F8].

The closed subschemes $L_m$ and $C_m$ are disjoint: on $U_0^{(m)}$ the line is $V(u_{02},u_{03})$ while $C_m\cap U_0^{(m)}=\mathrm{\varnothing }$; on $U_1^{(m)}$ a common point of $V(u_{12},u_{13})$ and $V(u_{10},1+u_{12}{u}_{13})$ would have $u_{12}=u_{13}=0$ and hence $1+u_{12}{u}_{13}=1\mathrm{\ne }0$, which is impossible; on $U_2^{(m)}$ and $U_3^{(m)}$ the line is empty. Hence $\mathrm{\mid }{L}_m\mathrm{\mid }\cap \mathrm{\mid }{C}_m\mathrm{\mid }=\mathrm{\varnothing }$ for $m=1,2$.

Let $I_i^L,I_i^C\subseteq A_i$ be the ideals generated by the dehomogenizations of $(x_2,x_3)$ and of $(x_0,x_1^2+x_2{x}_3)$ at the chart $i$ as displayed in steps 2.1 and 2.2; by step 1.1 their localizations correspond on overlaps, and the product ideals $I_i^L{I}_i^C$ have corresponding localizations as well, so the closed subschemes $\mathrm{Spec}(A_i\mathrm{/}{I}_i^L{I}_i^C)$ glue by [F2] to a closed subscheme $Z_m\subseteq X_m$ whose chart piece over $U_i^{(m)}$ is $\mathrm{Spec}(A_i\mathrm{/}{I}_i^L{I}_i^C)$, the closedness following from [F3]. On each chart the two ideals are comaximal: $I_0^C=I_2^L=I_3^L=A_i$, and on $U_1^{(m)}$ one has $(1+u_{12}{u}_{13})-u_{12}{u}_{13}=1$ in $I_1^L+I_1^C$. Hence [F5] gives, compatibly with the transitions of step 1.1, canonical isomorphisms $Z_m\cap U_i^{(m)}\cong (L_m\cap U_i^{(m)})\bigsqcup (C_m\cap U_i^{(m)}),$ which glue to a $k$-isomorphism $Z_m\cong L_m\bigsqcup C_m$; in particular $Z_m$ is a closed subscheme of $X_m$ whose closed subsets $L_m,C_m$ are disjoint and cover it.

Let $\sigma :Z_1\to Z_2$ be the isomorphism which on the component $L_1$ of $Z_1\cong L_1\bigsqcup C_1$ is ${\phi }_2\circ {\psi }_1:L_1\to C_2$ and on the component $C_1$ is ${\psi }_2^{-1}\circ {\phi }_1^{-1}:C_1\to L_2$; this is a $k$-isomorphism onto $Z_2\cong C_2\bigsqcup L_2$ with $\sigma (L_1)=C_2$ and $\sigma (C_1)=L_2$, and the source components are as in step 3.2.

The closed subschemes $L_m,C_m\subseteq X_m$ of steps 2.1 and 2.2 are disjoint by step 3.1, their disjoint union is the closed subscheme $Z_m$ by step 3.2, and $\sigma$ of step 4.1 exchanges them as required; the field $k$ is algebraically closed and $C_m$ is a nonsingular plane conic by step 2.2. So [F7] applies to the data $(X_1,X_2,L_m,C_m,\sigma )$: the closed-subscheme pushout $X=X_1{⨿}_Z{X}_2$, $Z:=Z_1$, exists as a $k$-scheme, the structure morphisms $a:X_1\to X$, $b:X_2\to X$ are closed immersions exhibiting each $X_m$ as a closed subscheme of $X$, and $X\to \mathrm{Spec}k$ is proper in the sense of [F11]. This proves the existence, closedness and properness clauses of the statement.

Suppose now that $h:X\to {\mathbb{P}}_k^N$ is a closed immersion over $\mathrm{Spec}k$ for some $N\ge 0$, so that $X\to \mathrm{Spec}k$ is projective in the sense of [F11]. Then $\mathcal{L}:=h^{\ast }{\mathcal{O}}_{{\mathbb{P}}^N}(1)$ is an invertible sheaf on $X$: $\mathcal{O}(1)$ is invertible by its gluing definition in [F8], and pullback preserves invertibility, since by the definition [F10] the pullback of a free rank-one module is free of rank one on the preimage of a trivializing open set. Hence $M_1:=a^{\ast }\mathcal{L}$ and $M_2:=b^{\ast }\mathcal{L}$ are invertible sheaves on $X_1,X_2\cong {\mathbb{P}}_k^3$. The composite $h\circ a:{\mathbb{P}}_k^3\to {\mathbb{P}}_k^N$ is again a closed immersion: over an affine open $V$ of ${\mathbb{P}}_k^N$ the preimage $h^{-1}(V)=\mathrm{Spec}(A\mathrm{/}I)$ is affine by [F3], the preimage $(ha{)}^{-1}(V)$ is a closed subscheme of it because $a$ is a closed immersion and closedness is local on the target by [F3], and a composite of closed subscheme inclusions is a closed immersion; [F3] then gives the composite closedness over the chosen affine cover of ${\mathbb{P}}_k^N$. Since $M_1\cong (ha{)}^{\ast }\mathcal{O}(1)$ is the pullback of $\mathcal{O}(1)$ along a closed immersion, the classification and positivity clauses of [F8] give a unique $n_1\in \mathbb{Z}$ with $M_1\cong {\mathcal{O}}_{{\mathbb{P}}^3}(n_1)$ and $n_1>0$; the same argument gives $M_2\cong {\mathcal{O}}_{{\mathbb{P}}^3}(n_2)$ with $n_2>0$.

Let $j_1:Z\to X_1$ and $j_2:Z\to X_2$ be the closed immersions used to glue (so $j_2=z_2\sigma$ in the notation of [F7], and $a{j}_1=b{j}_2$ because the pushout square commutes). Pulling $\mathcal{L}$ back along these two morphisms gives the same sheaf: $j_1^{\ast }{M}_1\cong (a{j}_1{)}^{\ast }\mathcal{L}=(b{j}_2{)}^{\ast }\mathcal{L}\cong j_2^{\ast }{M}_2$, the outer isomorphisms being the composition compatibility of pullback, which by [F10] is the canonical identification of stalks $({\mathcal{O}}_{Z,z}{\otimes }_{{\mathcal{O}}_{X_1,j_1(z)}}{M}_{1,j_1(z)})\cong {\mathcal{O}}_{Z,z}{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{L}}_x$ for $x=a{j}_1(z)$. This identification is compatible with the decompositions $Z\cong L_1\bigsqcup C_1$ and $Z_2\cong C_2\bigsqcup L_2$ of step 3.2 and with $\sigma$: passing to the component $L_1$ of $Z$ it reads $M_1{\mathrm{\mid }}_{L_1}\cong (\sigma {\mathrm{\mid }}_{L_1}{)}^{\ast }(M_2{\mathrm{\mid }}_{C_2})$, and passing to the component $C_1$ it reads $M_1{\mathrm{\mid }}_{C_1}\cong (\sigma {\mathrm{\mid }}_{C_1}{)}^{\ast }(M_2{\mathrm{\mid }}_{L_2})$, where $\sigma {\mathrm{\mid }}_{L_1}=({\phi }_2\circ {\psi }_1){\mathrm{\mid }}_{L_1}$ and $\sigma {\mathrm{\mid }}_{C_1}=({\psi }_2^{-1}\circ {\phi }_1^{-1}){\mathrm{\mid }}_{C_1}$ by step 4.1.

Restrict to the component $L_1$. By the line clause of [F8] the pullback of $M_1{\mathrm{\mid }}_{L_1}$ along ${\psi }_1^{-1}:{\mathbb{P}}_k^1\to L_1$ is ${\mathcal{O}}_{{\mathbb{P}}^1}(n_1)$. On the other hand step 6.2 identifies $M_1{\mathrm{\mid }}_{L_1}$ with $(\sigma {\mathrm{\mid }}_{L_1}{)}^{\ast }(M_2{\mathrm{\mid }}_{C_2})$, so pulling back along ${\psi }_1^{-1}$ gives $(\sigma {\mathrm{\mid }}_{L_1}\circ {\psi }_1^{-1}{)}^{\ast }(M_2{\mathrm{\mid }}_{C_2})={\phi }_2^{\ast }(M_2{\mathrm{\mid }}_{C_2})$, which is ${\mathcal{O}}_{{\mathbb{P}}^1}(2n_2)$ by the conic clause of [F8] applied to the nonsingular plane conic $C_2$ of step 2.2 with the isomorphism ${\phi }_2$. Hence ${\mathcal{O}}_{{\mathbb{P}}^1}(n_1)\cong {\mathcal{O}}_{{\mathbb{P}}^1}(2n_2)$, and $n_1=2n_2$ by [F9].

Restrict to the component $C_1$. By the conic clause of [F8] the pullback of $M_1{\mathrm{\mid }}_{C_1}$ along ${\phi }_1:{\mathbb{P}}_k^1\to C_1$ is ${\mathcal{O}}_{{\mathbb{P}}^1}(2n_1)$. By step 6.2 the sheaf $M_1{\mathrm{\mid }}_{C_1}$ is $(\sigma {\mathrm{\mid }}_{C_1}{)}^{\ast }(M_2{\mathrm{\mid }}_{L_2})$, so pulling back along ${\phi }_1$ gives $(\sigma {\mathrm{\mid }}_{C_1}\circ {\phi }_1{)}^{\ast }(M_2{\mathrm{\mid }}_{L_2})={\psi }_2^{-1\ast }(M_2{\mathrm{\mid }}_{L_2})$, the pullback of $M_2{\mathrm{\mid }}_{L_2}$ along ${\psi }_2^{-1}:{\mathbb{P}}_k^1\to L_2$, which is ${\mathcal{O}}_{{\mathbb{P}}^1}(n_2)$ by the line clause of [F8]. Hence ${\mathcal{O}}_{{\mathbb{P}}^1}(2n_1)\cong {\mathcal{O}}_{{\mathbb{P}}^1}(n_2)$ and $2n_1=n_2$ by [F9].

Combining $n_1=2n_2$ of step 7.1 with $2n_1=n_2$ of step 7.2 gives $n_1=4n_1$, hence $3n_1=0$ and $n_1=0$ in $\mathbb{Z}$, contradicting $n_1>0$ from step 6.1. Therefore no closed immersion $X\to {\mathbb{P}}_k^N$ over $k$ exists for any $N\ge 0$, and by [F11] the structure morphism $X\to \mathrm{Spec}k$ is not projective, while it is proper by step 5.1: properness does not imply projectivity over an algebraically closed field. The Axiom of Choice [F12] is used exactly through the AC-declared suppliers [F3], [F6] and [F7] cited in steps 3.2 and 5.1 and in the closedness computation of step 6.1; all other steps make finitely many explicit choices of charts, coordinates and generators.