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110.66 Non-effective descent data for projective schemes

In the chapter on descent we have seen that descent data for schemes relative to an fpqc morphism are effective for several classes of morphisms. In particular, affine morphisms and more generally quasi-affine morphisms satisfy descent for fpqc coverings (Descent, Lemma 35.38.1). This is not true for projective morphisms.

Lemma 110.66.1. There is an etale covering $X\to S$ of schemes and a descent datum $(V/X,\varphi )$ relative to $X\to S$ such that $V\to X$ is projective, but the descent datum is not effective in the category of schemes.

Proof. We imitate Hironaka's example of a smooth separated complex algebraic space of dimension 3 which is not a scheme [Example B.3.4.2, H].

Consider the action of the group $G = \mathbf{Z}/2 = \{ 1, g\} $ on projective 3-space $\mathbf{P}^3$ over the complex numbers by

\[ g[x,y,z,w] = [y,x,w,z]. \]

The action is free outside the two disjoint lines $L_1=\{ [x,x,z,z]\} $ and $L_2=\{ [x,-x,z,-z]\} $ in ${\mathbf P}^3$. Let $Y={\mathbf P}^3-(L_1\cup L_2)$. There is a smooth quasi-projective scheme $S=Y/G$ over ${\mathbf C}$ such that $Y\to S$ is a $G$-torsor (Groupoids, Definition 39.11.3). Explicitly, we can define $S$ as the image of the open subset $Y$ in ${\mathbf P}^3$ under the morphism

\begin{align*} {\mathbf P}^3 & \to \text{Proj } {\mathbf C}[x,y,z,w]^ G\\ & = \text{Proj } {\mathbf C}[u_0,u_1,v_0,v_1,v_2]/(v_0v_1=v_2^2), \end{align*}

where $u_0=x+y$, $u_1=z+w$, $v_0=(x-y)^2$, $v_1=(z-w)^2$, and $v_2=(x-y)(z-w)$, and the ring is graded with $u_0,u_1$ in degree 1 and $v_0,v_1,v_2$ in degree 2.

Let $C=\{ [x,y,z,w]: xy=z^2, w=0\} $ and $D=\{ [x,y,z,w]: xy=w^2, z=0\} $. These are smooth conic curves in ${\mathbf P}^3$, contained in the $G$-invariant open subset $Y$, with $g(C)=D$. Also, $C\cap D$ consists of the two points $P:=[1,0,0,0]$ and $Q:=[0,1,0,0]$, and these two points are switched by the action of $G$.

Let $V_ Y\to Y$ be the scheme which over $Y-P$ is defined by blowing up $D$ and then the strict transform of $C$, and over $Y-Q$ is defined by blowing up $C$ and then the strict transform of $D$. (This is the same construction as in the proof of Lemma 110.65.1, except that $Y$ here denotes an open subset of ${\mathbf P}^3$ rather than all of ${\mathbf P}^3$.) Then the action of $G$ on $Y$ lifts to an action of $G$ on $V_ Y$, which switches the inverse images of $Y-P$ and $Y-Q$. This action of $G$ on $V_ Y$ gives a descent datum $(V_ Y/Y,\varphi _ Y)$ on $V_ Y$ relative to the $G$-torsor $Y\to S$. The morphism $V_ Y\to Y$ is proper but not projective, as shown in the proof of Lemma 110.65.1.

Let $X$ be the disjoint union of the open subsets $Y-P$ and $Y-Q$; then we have surjective etale morphisms $X\to Y\to S$. Let $V$ be the pullback of $V_ Y\to Y$ to $X$; then the morphism $V\to X$ is projective, since $V_ Y\to Y$ is a blowup over each of the open subsets $Y-P$ and $Y-Q$. Moreover, the descent datum $(V_ Y/Y,\varphi _ Y)$ pulls back to a descent datum $(V/X,\varphi )$ relative to the etale covering $X\to S$.

Suppose that this descent datum is effective in the category of schemes. That is, there is a scheme $U\to S$ which pulls back to the morphism $V\to X$ together with its descent datum. Then $U$ would be the quotient of $V_ Y$ by its $G$-action.

\[ \xymatrix{ V \ar[r]\ar[d]& X\ar[d] \\ V_ Y \ar[r]\ar[d]& Y\ar[d] \\ U \ar[r]& S } \]

Let $E$ be the inverse image of $C\cup D\subset Y$ in $V_ Y$; thus $E\rightarrow C\cup D$ is a proper morphism, with fibers isomorphic to ${\mathbf P}^1$ over $(C\cup D)-\{ P,Q\} $. The inverse image of $P$ in $E$ is a union of two lines $L_0$ and $M_0$. It follows that the inverse image of $Q=g(P)$ in $E$ is the union of two lines $L_0'=g(M_0)$ and $M_0'=g(L_0)$. As shown in the proof of Lemma 110.65.1, we have a rational equivalence $L_0+M_0'=L_0+g(L_0)\sim 0$ on $E$.

By descent of closed subschemes, there is a curve $L_1\subset U$ (isomorphic to ${\mathbf P}^1$) whose inverse image in $V_ Y$ is $L_0\cup g(L_0)$. (Use Descent, Lemma 35.37.1, noting that a closed immersion is an affine morphism.) Let $R$ be a complex point of $L_1$. Since we assumed that $U$ is a scheme, we can choose a function $f$ in the local ring $O_{U,R}$ that vanishes at $R$ but not on the whole curve $L_1$. Let $D_{\text{loc}}$ be an irreducible component of the closed subset $\{ f = 0\} $ in $\mathop{\mathrm{Spec}}O_{U,R}$; then $D_{\text{loc}}$ has codimension 1. The closure of $D_{\text{loc}}$ in $U$ is an irreducible divisor $D_ U$ in $U$ which contains the point $R$ but not the whole curve $L_1$. The inverse image of $D_ U$ in $V_ Y$ is an effective divisor $D$ which intersects $L_0\cup g(L_0)$ but does not contain either curve $L_0$ or $g(L_0)$.

Since the complex 3-fold $V_ Y$ is smooth, $O(D)$ is a line bundle on $V_ Y$. We use here that a regular local ring is factorial, or in other words is a UFD, see More on Algebra, Lemma 15.121.2. The restriction of $O(D)$ to the proper surface $E\subset V_ Y$ is a line bundle which has positive degree on the 1-cycle $L_0+g(L_0)$, by our information on $D$. Since $L_0+g(L_0)\sim 0$ on $E$, this contradicts that the degree of a line bundle is well-defined on 1-cycles modulo rational equivalence on a proper scheme over a field (Chow Homology, Lemma 42.20.3 and Lemma 42.28.2). Therefore the descent datum $(V/X,\varphi )$ is in fact not effective; that is, $U$ does not exist as a scheme. $\square$

In this example, the descent datum is effective in the category of algebraic spaces. More precisely, $U$ exists as a smooth separated algebraic space of dimension 3 over ${\mathbf C}$, for example by Algebraic Spaces, Lemma 65.14.3. Hironaka's 3-fold $U$ is a small resolution of the blowup $S'$ of the smooth quasi-projective 3-fold $S$ along the irreducible nodal curve $(C\cup D)/G$; the 3-fold $S'$ has a node singularity. The other small resolution of $S'$ (differing from $U$ by a “flop”) is again an algebraic space which is not a scheme.

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