Éléments de Géométrie Algébrique — English

§V.6. Invertible sheaves and divisors relative to projective and multiprojective fibrations; linear systems of divisors (formerly EGA IV §21)

This section was originally drafted as §21 of EGA IV, then re-allocated to EGA V (Chapter V §6) without ever being published in either place. We lead with the §V numbering and attach (formerly IV, 21) parenthetically at the first occurrence of each cross-reference into the old numbering; subsequent in-section cross-references use V numbering alone.

The four subsections, following the prenote's own table of contents, are:

1. Invertible sheaves on projective and multiprojective fibrations.
2. Representability of ${\mathrm{Div}}_{X/S}^L$: relative divisors on projective and multiprojective fibrations.
3. Linear systems of divisors and morphisms into projective fibrations.
4. Linear systems of divisors and invertible modules.

The results developed here, and in the following section, are already partly global in nature; they give complements about projective schemes (II, III) and pure local results of the present Chapter V. One of the aims of the paragraph is to develop the language of linear systems of divisors, which is connected on the one hand to the classification of morphisms into a projective fibration, and on the other to the classification of invertible modules over a given prescheme.

The "parameter schemes" really natural for the linear systems of divisors are the Brauer-Severi schemes, which generalize projective fibrations and may be defined, for example, as those fibrations which become isomorphic to a projective fibration after an étale surjective base extension. Since their study uses descent theory (taken up in Chapter V¹ of the original design), and since their classification is equivalent to the classification of torsors under the projective group, we postpone the study of such schemes — and of their connection with the notion of linear system — to Chapter VI.

From a technical point of view, the main result of the section is Theorem (1.1), which determines the Picard group of a projective fibration in terms of that of the base, together with its immediate corollaries in subsections 1 and 2.

V.6.1. Invertible sheaves on projective fibrations; automorphisms

Editorial note (Blass). The bulk of this subsection was crossed out in the manuscript up to the point we shall indicate below; we restore it in full.

This subsection treats the determination of invertible sheaves on a projective fibration and its application to the automorphisms of such a fibration.

Theorem (1.1).

Let $S$ be a prescheme, $E$ a locally free ${\mathcal{O}}_S$-module of finite rank $\ge 2$ at every point, and $P=\mathbb{P}(E)$ the projective fibration that it defines, with structural morphism $f:P\to S$. Then for every invertible module $L$ over $P$ one can find a family $(S_n{)}_{n\in \mathbb{Z}}$ of disjoint open subsets of $S$ covering $S$, indexed by $\mathbb{Z}$, and an invertible module $M$ over $S$, such that the restriction of $L$ to $f^{-1}(S_n)$ is isomorphic to that of

$$M\otimes {\mathcal{O}}_P(n)=f\ast (M)(n).$$

Moreover, the family $(S_n)$ is uniquely determined by these conditions, and $M$ is then determined up to a unique isomorphism.

Remark (1.2).

If we drop the assumption on the rank of $E$, then $S$ decomposes canonically into the sum-prescheme of three open subsets $S^0$, $S^1$, $S^2$ over which the rank of $E$ is respectively 0, 1, and $\ge 2$. The determination of the invertible modules over $P$ reduces to that over $f^{-1}(S^i)$ for $i=0,1,2$. The case $i=2$ follows from (1.1); on the other hand, $f^{-1}(S^1)$ is $S^1$-isomorphic to $S^1$, so its Picard group is just $\mathrm{Pic}(S^1)$; and finally $f^{-1}(S^0)$ is empty, so its Picard group is zero.

Corollary (1.3).

Under the assumptions of (1.1), suppose moreover that $S$ is connected and non-empty. Then every invertible module $L$ over $P$ is isomorphic to a module of the form $f\ast (M)(n)$, where $n\in \mathbb{Z}$ and $M$ is an invertible module over $S$. Furthermore $n$ is uniquely determined, and $M$ is determined up to a unique isomorphism, by the data of $L$.

Another way to formulate this corollary is the following. Consider the natural homomorphisms $\mathrm{Pic}(S)\to \mathrm{Pic}(P)$ (deduced from $f:P\to S$) and $\mathbb{Z}\to \mathrm{Pic}(P)$ (determined by the class $[{\mathcal{O}}_P(1)]\in \mathrm{Pic}(P)$). We deduce a canonical homomorphism

$$(\ast )\mathrm{Pic}(S)\times \mathbb{Z}⟶\mathrm{Pic}(P),$$

defined in any case, without any restrictive assumption on $S$ or on $E$. This gives:

Corollary (1.4).

Under the conditions of (1.1), if $S\ne \varnothing$, then the homomorphism (*) is injective; it is bijective if $S$ is also connected.

If we drop the assumption that the rank of $E$ is $\ge 2$, it follows from (1.2) and (1.4) that the homomorphism (*) is still surjective when $S$ is connected, but not necessarily injective: its kernel is isomorphic to $\mathbb{Z}$ if $E$ is of rank 1, and to $\mathrm{Pic}(S)\times \mathbb{Z}$ if $E$ is of rank zero.

Proof of (1.1)

We start from uniqueness. Suppose first $S=\mathrm{Spec}(K)$ is the spectrum of a field. Then ${\mathcal{O}}_P(n)$ is not isomorphic to ${\mathcal{O}}_P(m)$ for $n\ne m$; equivalently, ${\mathcal{O}}_P(m-n)$ is not isomorphic to ${\mathcal{O}}_P$ unless $m-n=0$. Indeed, this follows from the fact that ${\mathcal{O}}_P(1)$ is ample and $\dim P\ge 1$ (which uses $rankE\ge 2$): we may suppose $d=m-n\ge 0$; if $d>0$, then ${\mathcal{O}}_P(d)$ would be ample, so it could be isomorphic to ${\mathcal{O}}_P$ only if $P$ were quasi-affine, hence finite (since it is proper over $K$) — contradicting $\dim P\ge 1$. This already proves the uniqueness of the family $(S_n)$.

For the uniqueness of $M$ up to a unique isomorphism, we are reduced to the case $S=S_n$. In this case the isomorphism

$$(\ast \ast )M\stackrel{\sim }{\to }{f}_{\ast }(L(-n))$$

is uniquely determined by the isomorphism $L\stackrel{\sim }{\to }f\ast (M)(n)$. Indeed, the latter defines an isomorphism $L(-n)\stackrel{\sim }{\to }f\ast (M)$, which yields an isomorphism of the right-hand side of (**) with $f_{\ast }(f\ast (M))\cong M\otimes f_{\ast }({\mathcal{O}}_P)$, and since $f_{\ast }({\mathcal{O}}_P)\stackrel{\sim }{\to }{\mathcal{O}}_S$, we deduce (**).

We now prove existence of the $(S_n)$ and $M$. By the uniqueness already shown the question is local over $S$, so we are reduced to the following corollary:

Corollary (1.5).

Let $E$ be locally free of finite rank over $S$, $P=\mathbb{P}(E)$, $L$ an invertible module over $P$, and $s\in S$. Then there exists an open neighbourhood $U$ of $s$ and an integer $n\in \mathbb{Z}$ such that $L|f^{-1}(U)$ is isomorphic to ${\mathcal{O}}_P(n)|f^{-1}(U)$.

Of course, since the rank of $E$ at $s$ is $\ge 2$, the integer $n$ is well defined.

Moreover, (1.5) is trivial when the rank of $E$ at $s$ is $\le 1$. So we may assume that $E$ is locally free of constant rank $\ge 2$; the question being local, we may suppose $E={\mathcal{O}}_S^{r+1}$ and hence $P={\mathbb{P}}_S^r$.² By the standard brief procedure of (IV, §8), we are also reduced to the case where $S$ is noetherian. We proceed in two steps:

(a) Suppose $S=\mathrm{Spec}(K)$ is the spectrum of a field. Then $L$ is defined by a graded module of finite type $\mathcal{L}$ over the graded ring $K[t_0,\cdots ,t_r]$. The restriction $L^{\prime }$ of $L$ to the punctured affine space ${\mathbb{E}}_K^{r+1}-\{0\}$ is the inverse image of $L$ under the canonical projection morphism $q:{\mathbb{E}}_K^{r+1}-\{0\}\to {\mathbb{P}}_K^r$, and is therefore an invertible module. Let $i:{\mathbb{E}}_K^{r+1}-\{0\}\hookrightarrow {\mathbb{E}}_K^{r+1}$ be the canonical immersion. The affine ring $K[t_0,\cdots ,t_r]$ of ${\mathbb{E}}_K^{r+1}$ is factorial; a fortiori its localization at the origin is factorial, and that ring has dimension $\ge 2$. It follows that $i_{\ast }(L^{\prime })$ is an invertible module, corresponding to an invertible graded module $\mathcal{M}$ over $K[t_0,\cdots ,t_r]$.³ Moreover $\mathcal{M}=\Gamma ({\mathbb{E}}_K^{r+1}-\{0\},q\ast (\mathcal{L}))$ is graded in a natural fashion, and the homomorphism $\mathcal{L}\to \mathcal{M}$ is an isomorphism at every point of ${\mathbb{E}}_K^{r+1}$ distinct from the origin.

Replacing $\mathcal{L}$ by $\mathcal{M}$, we are reduced to the case of an invertible $\mathcal{L}$. Since $K[t_0,\cdots ,t_r]$ is factorial, $\mathcal{L}$ is then free of rank one when we ignore its grading. A standard lemma — without doubt available in Bourbaki — implies that it is also free of rank one as a graded module, which means that the associated ${\mathcal{O}}_P$-module is isomorphic to some ${\mathcal{O}}_P(n)$. From the editorial point of view, it would be clumsy to begin by considering an $\mathcal{L}$ of finite type; one should rather begin by introducing directly $\mathcal{L}=\Gamma ({\mathbb{E}}_K^{r+1}-\{0\},q\ast (L))$, define the module $\mathcal{L}=q_{\ast }(\mathcal{L})$, and remark (by Chapter II) that $\mathcal{L}$ is a graded module that determines $L$; then one shows that $\mathcal{L}$ is free of rank 1 as a graded module by the indicated reasoning.

(b) General case. This is deduced from case (a) using (III, 4.6.5) together with the vanishing relation $H^1({\mathbb{P}}_K^r,{\mathcal{O}}_{{\mathbb{P}}_K^r})=0$ established in (III, §2). q.e.d.

A variant of (1.4)

Let $\mathbb{Z}(S)$ denote the set of locally constant functions $S\to \mathbb{Z}$. We define an evident homomorphism $\mathbb{Z}(S)\to \mathrm{Pic}(P)$: an element $n\in \mathbb{Z}(S)$ corresponds to a partition $(S_n{)}_{n\in \mathbb{Z}}$ of $S$ into disjoint open subsets (some possibly empty), and to such a partition we associate the invertible module ${\mathcal{O}}_P(n)$ whose restriction to $f^{-1}(S_n)$ is ${\mathcal{O}}_P(n)$. We thus obtain a variant

  (* bis)          Pic(S) × ℤ(S) ⟶ Pic(P),

and a statement more general and more satisfactory than (1.4): under the conditions of (1.1), $(\ast bis)$ is a bijection. (Indeed, when $S\ne \varnothing$, the canonical map $\mathbb{Z}\to \mathbb{Z}(S)$ sending $n$ to the constant function of value $n$ is injective; it is bijective if and only if $S$ is connected. So we recover (1.4) formally, and the statement above is the most general form.)

We also remark — and this is a remark worth making — that in the language of the Picard scheme (which will be introduced in Chapter V.1.1⁴), the preceding statement reads simply: under the conditions of (1.1), the canonical homomorphism ${\mathbb{Z}}_S\to {\mathrm{Pic}}_{P/S}$ of constant group schemes $\mathbb{Z}$ over $S$ into the relative Picard scheme, deduced from the section of the latter defined by ${\mathcal{O}}_P(1)$, is an isomorphism.

Morphisms between projective fibrations; degree

Let $P$ be a projective fibration over a field $K$. An invertible module $L$ over $P$ is said to be of degree $n$ if $L$ is isomorphic to ${\mathcal{O}}_P(n)$; if $\dim P\ge 1$, this determines $n$ in terms of $L$, but if $\dim P\le 0$ ($P$ empty or reduced to a point) then $L$ is of degree $n$ for every $n$. By (1.1) and (1.2), to say that $L$ is of degree $n$ amounts to saying that the class of $L$ in $\mathrm{Pic}(P)$ lies in the image of $\mathrm{Pic}(S)\times \{n\}$ under the homomorphism (*), i.e. that $L$ is isomorphic to a module of the form $f\ast (M)(n)$ with $M$ invertible over $S$. Moreover, if the fibres of $P$ are non-empty ($E$ everywhere of rank $\ge 1$), then $M$ is determined up to a unique isomorphism by $L$, again by (1.1) and (1.2).

Grothendieck note. I notice on this occasion that it would be proper to announce (1.1) without any hypothesis on the rank of $E$: every invertible $L$ over $P$ can be taken in the form indicated in that statement, and if the fibres of $P$ are non-empty (i.e. the rank of $E$ is $\ge 1$), then the partition of $S$ is also uniquely determined by the choice of $L$. In that case Remark (1.2) is absorbed into the proof and disappears.

Let $P^1=\mathbb{P}(E^1)$ be a second projective fibration. The determination of $\mathrm{Pic}(P)$ allows us in principle to determine the $S$-morphisms $g:P\to P^1$: these are defined by an invertible module $L=g\ast ({\mathcal{O}}_{P^1}(1))$ over $P$ together with a homomorphism $E^1\to f_{\ast }(L)$ (such that the associated homomorphism $f\ast (E^1)\to L$ is surjective), modulo an isomorphism of $L$. We say that $g:P\to P^1$ is of degree $n$ if $L=g\ast ({\mathcal{O}}_{P^1}(1))$ is of degree $n$. It is enough, evidently, to determine the homomorphisms of degree $n$ for each given $n$.

Note that if $g$ is of degree $n$ and $P$ has fibres of dimension $\ge 1$, then necessarily $n\ge 0$ (since over a field $K$ with $\dim P\ge 1$, ${\mathcal{O}}_P(n)$ is generated by its sections only when $n\ge 0$). One could of course restrict to the case where the fibres of $P$ have dimension $\ge 1$, by proceeding as in (1.2). Since

  f_*(f*(M)(n)) = M ⊗ f_*(𝒪_P(n)) = M ⊗ Sym^n(E),

the determination of $S$-morphisms $g:P\to P^1$ reduces to the determination, up to isomorphism, of pairs $(M,u)$, where $M$ is an invertible module over $S$ and $u:E^1\to M\otimes Sy{m}^n(E)$ is a homomorphism such that the corresponding homomorphism $f\ast (E^1)\to f\ast (M)(n)$ is an epimorphism. The morphism $g$ then determines a first global invariant over $S$, namely the class $[M]\in \mathrm{Pic}(S)$; with $M$ fixed, the corresponding $g$ correspond to a certain subset of the quotient set $\mathrm{Hom}(E^1,M\otimes Sy{m}^n(E))/\Gamma (S,{\mathcal{O}}_S{)}^{\times }$, the passage to the quotient by $\Gamma (S,{\mathcal{O}}_S{)}^{\times }$ corresponding to "module isomorphism" in the description of $g:P\to P^1$ via invertible modules.

Nota bene. The endomorphisms (resp. automorphisms) of an invertible $L$ over a projective fibration $P$ correspond to sections (resp. invertible sections) of ${\mathcal{O}}_P$ over $P$, or — what amounts to the same — to sections (resp. invertible sections) of ${\mathcal{O}}_S\stackrel{\sim }{\to }{f}_{\ast }({\mathcal{O}}_P)$ over $S$.

Special cases

(i) $n=0$. We take the homomorphisms $E^1\to M$ that are surjective, i.e. everywhere non-zero, modulo isomorphism of $M$. We find exactly the morphisms $g:P\to P^1$ of the form induced by a section $h$ of $P^1$ over $S$ (namely those determined by the invertible quotient $M$ of $E^1$). Thus the $S$-morphisms of degree 0 from $P$ into $P^1$ are the constant morphisms relative to $S$.

End of crossed-out portion (per the manuscript marker).

(ii) $n=1$. We take the homomorphisms $E^1\to E\otimes M$ that are surjective — as one verifies immediately. The corresponding homomorphism $g:P\to P^1$ is then the composition

  ℙ(E) ⥲ ℙ(E ⊗ M) ↪ ℙ(E^1),

where the first map is the canonical isomorphism described in Chapter II and the second is the canonical closed immersion deduced from the epimorphism $E^1\to E\otimes M$. We call the homomorphisms $P\to P^1$ of this form linear; the morphisms $g:P\to P^1$ of degree 1 are then exactly the linear ones.

To finish, we determine the isomorphisms of $P$ with $P^1$.

Theorem (1.6).

Let $S$ be a prescheme, and let $P=\mathbb{P}(E)$, $P^1=\mathbb{P}(E^1)$ be two projective fibrations over $S$ defined by $E$, $E^1$ locally free of finite type. Then every $S$-isomorphism $g:P\to P^1$ is definable as a composition

  ℙ(E) ⥲ ℙ(E ⊗ M) ⥲ ℙ(E^1),

where $M$ is an invertible module over $S$, the first map is the canonical isomorphism of Chapter II, and the second is the isomorphism deduced from an isomorphism $u:E^1\stackrel{\sim }{\to }E\otimes M$. Provided the fibres of $P$ are non-empty (resp. of dimension $\ge 1$), $M$ (resp. the pair $(M,u)$) is determined up to a unique isomorphism by $g$.

By the foregoing considerations we are reduced to showing that $g$ is of degree one. This in turn reduces (by passing to fibres) to the case where $S$ is the spectrum of a field and (without loss of generality) $\dim P\ge 1$. But then ${\mathcal{O}}_P(1)$ is intrinsically characterized — independently of the way $P$ is realized as a projective fibration — as the ample generator of $\mathrm{Pic}(P)$ (between the two generators ${\mathcal{O}}_P(1)$ and ${\mathcal{O}}_P(-1)$); consequently, if $g:P\to P^1$ is an isomorphism, then $g\ast ({\mathcal{O}}_{P^1}(1))$ is ample and is therefore isomorphic to ${\mathcal{O}}_P(1)$. We are done.

In a less elaborate local form, we may state:

Corollary (1.7).

Under the conditions of (1.6), every $S$-isomorphism $g:P\to P^1$ can be described, in a neighbourhood $U$ of each $s\in S$, by an isomorphism $u:E|U\stackrel{\sim }{\to }{E}^1|U$, the latter being well defined modulo multiplication by an element of $\Gamma (U,{\mathcal{O}}_U{)}^{\times }$.

In particular:

Corollary (1.8).

Let $S$ be a prescheme, $P=\mathbb{P}(E)$ a projective fibration over $S$ defined by $E$ locally free of finite type, and $u$ an automorphism of $P$. Then $u$ is determined, in a neighbourhood $U$ of every point $s\in S$, by an automorphism ũ of $E|U$, the latter being well defined by $u$ modulo multiplication by an element of $\Gamma (U,{\mathcal{O}}_U{)}^{\times }$.

Remark (1.9).

From (1.8) one deduces easily that the group functor ${\mathrm{Aut}}_S(P)$ over $S$ is representable by an affine $S$-prescheme of finite presentation, which may also be interpreted as the quotient group-scheme of the linear group-scheme $GL(E)$ by its centre ${\mathbb{G}}_m$. This group-prescheme is called the prescheme of projective groups, or simply the projective group, defined by $E$, and is denoted $PGL(E)$. If $E$ is free, $E\cong {\mathcal{O}}_S^{r+1}$, then $PGL(E)$ is just the group-prescheme $PGL(r{)}_S$ deduced by base change $S\to \mathrm{Spec}(\mathbb{Z})$ from the analogous group-scheme $PGL(r)$ over $\mathrm{Spec}(\mathbb{Z})$, called the absolute projective group.

Grothendieck note. End of Appendix 1. A marginal remark next to Remark (1.9) is partly illegible: the legible fragment refers to $\mathbb{P}(E\otimes E^{\vee })$ and the "open set defined by the non-vanishing of the determinant", presumably an embedded description of $PGL(E)$ as the open subscheme of $\mathbb{P}(\mathrm{End}(E))$ cut out by $det\ne 0$.⁵

V.6.2. Relative divisors and invertible sheaves on projective and multiprojective fibrations

(2.1). Let, as in V.6.1, $P=\mathbb{P}(E)$, with $E$ locally free over $S$ of rank $\ge 2$ everywhere. We propose to determine the set ${\mathrm{Div}}^{+}(P/S)$ of positive relative divisors over $P$ with respect to $S$. Such a divisor is the same data as an invertible module $L$ over $P$ together with a transversally regular section $\varphi$ of $L$. By (1.1), ignoring a possible partition of $S$ if $S$ is not connected, $L$ is isomorphic to $M\otimes {\mathcal{O}}_P(n)$ for some invertible $M$ over $S$, determined up to a unique isomorphism by $L$. Furthermore, by the canonical isomorphisms

  (*)            f_*(L) ≅ M ⊗ f_*(𝒪_P(n)) ≅ M ⊗ Sym^n(E),

to give a section $\varphi$ of $L$ amounts to giving a section $\psi$ of $M\otimes Sy{m}^n(E)$. Since the fibres of $P/S$ are integral, $\varphi$ is transversally regular (i.e. regular on each fibre) if and only if $\psi (s)\ne 0$ for every $s\in S$; equivalently, the homomorphism $M^{\vee }\to Sy{m}^n(E)$ defined by $\psi$ is "universally injective" (i.e. locally an isomorphism onto a direct factor); equivalently again, its transpose $Sy{m}^n(E{)}^{\vee }\to M$ is surjective.

We say that a relative divisor $D$ over $P$ is of degree $n$ if ${\mathcal{O}}_P(D)=L$ is of degree $n$ in the sense of V.6.1. Since $D\ge 0$, this forces $n\ge 0$: if we had $n<0$, every section of $L$ over fibres would vanish.⁶ By (1.1), if $D$ is a positive relative divisor over $P$, then there is a unique decomposition of $S$ into a sum of disjoint open subsets $(S_n{)}_{n\in \mathbb{N}}$ such that for each $n$, $L|P/S_n$ is of degree $n$. This reduces the determination of the set of positive relative divisors to the determination, for given $n$, of those of degree $n$.⁷

The foregoing reflections give:

Proposition (2.2).

Under the above hypotheses, there is a one-to-one correspondence between the set ${\mathrm{Div}}^n(P/S)$ of positive relative divisors of degree $n$ over $P$ and the set of invertible quotient modules $M$ of $Sy{m}^n(E{)}^{\vee }$ (equivalently, of invertible submodules of $Sy{m}^n(E)$ which are locally direct factors $M^{\vee }$). If $D$ and $M$ correspond, then ${\mathcal{O}}_P(D)$ is canonically isomorphic to $M\otimes {\mathcal{O}}_P(n)$, and the canonical section $s_D$ is identified, under this isomorphism, with the section given by (*).

Note that this description is compatible with base change in $S$. Taking into account the interpretation of invertible quotient modules of $Sy{m}^n(E{)}^{\vee }$ as sections over $S$ of $\mathbb{P}(Sy{m}^n(E))$, we find:

Corollary (2.3).

The subfunctor ${\mathrm{Div}}_{P/S}^n$ of ${\mathrm{Div}}_{P/S}^{+}$ is canonically representable by $\mathbb{P}(Sy{m}^n(E{)}^{\vee })$.

Taking into account the reduction of V.6.2.1, it follows that:

Corollary (2.4).

The functor ${\mathrm{Div}}_{P/S}^{+}$ is canonically representable by the sum-prescheme of the $\mathbb{P}(Sy{m}^n(E{)}^{\vee })$ for $n\in \mathbb{N}$.⁸

The multiprojective case

(2.5). Now suppose we are given a finite family $(E_i{)}_{i\in I}$ of locally free modules over $S$, with $P_i=\mathbb{P}(E_i)$ and $P={\prod }_{i\in I,S}{P}_i$ the multiprojective fibration over $S$ defined by the $(E_i)$. For brevity we write ${\mathcal{O}}_i(n)$ for the inverse image to $P$ of the invertible module ${\mathcal{O}}_{P_i}(n)$ over $P_i$. For every system of integers $n=(n_i{)}_{i\in I}\in {\mathbb{Z}}^I$, we set

  𝒪_P(n) = ⨂_{i ∈ I} 𝒪_i(n_i) = ⨂_{i ∈ I, 𝒪_S} 𝒪_{P_i}(n_i).

With this notation, (1.1) generalizes as follows:

Proposition (2.6).

Suppose that the $E_i$ have rank $\ge 2$ everywhere. Then for every invertible module $L$ over the multiprojective fibration $P$, there exists a decomposition of $S$ into a sum of disjoint open subsets $(S_n{)}_{n\in {\mathbb{Z}}^I}$ and an invertible module $M$ over $S$ such that

  L | P/S_n ≅ M ⊗ 𝒪_P(n) | P/S_n.

Moreover, the $S_n$ are uniquely determined, and $M$ is determined up to a unique isomorphism.

The proof consists of an immediate reduction to (1.1). Under the conditions of (2.6), we may therefore associate to every invertible $L$ over $P$ a multidegree $n=(n_i{)}_{i\in I}\in \mathbb{Z}(S{)}^I$, which characterizes $L$ up to a unique isomorphism provided $\mathrm{Pic}(S)=0$. The $n_i$ (called the partial degree of $L$ with respect to the factor $P_i$ of index $i$) admit the following interpretation: if for each $i$ we choose a section $g_i$ of $P_i$ over $S$ (such sections exist locally over $S$ in any case), the system defines, for each $i$, an $S$-morphism $g_i:P_i\to P$; we then have

$$n_i=deg{g}_i^{\ast }(L),$$

it being understood that in general $n_i$ is not an integer but a locally constant function $S\to \mathbb{Z}$.

Proceeding as in V.6.1, we may deduce from (2.6) the determination of morphisms of one multiprojective fibration into another, and in particular the determination of automorphisms of multiprojective fibrations. More interesting for us, in view of (V, 5) on the resultant and discriminant of forms, will be the determination of positive relative divisors on a multiprojective fibration.

(2.7). If $D$ is a relative divisor over $P$, we define its multidegree as that of ${\mathcal{O}}_P(D)$. As above, the determination of ${\mathrm{Div}}^{+}(P/S)$ reduces to that of ${\mathrm{Div}}^n(P/S)$ for given multidegree $n\in {\mathbb{Z}}^I$. This gives an isomorphism $L={\mathcal{O}}_P(D)\cong M{\otimes }_{{\mathcal{O}}_S}{\mathcal{O}}_P(n)$, and by (II, §2)

  (**)           f_*(L) = M ⊗ f_*(𝒪_P(n)) = M ⊗ ⨂_i Sym^{n_i}(E_i).

Proceeding now as in (2.2), we find:

Proposition (2.8).

With the preceding notations, there is a one-to-one canonical correspondence between the set ${\mathrm{Div}}^n(P/S)$ of positive relative divisors of multidegree $n$ over $P$ and the set of invertible quotient modules $M$ of $({⨂}_{i}Sy{m}^{n_i}(E_i){)}^{\vee }$ (equivalently, of invertible submodules locally direct factors $M^{\vee }$ of ${⨂}_{i}Sy{m}^{n_i}(E_i)$). If $D$ and $M$ correspond, then ${\mathcal{O}}_P(D)$ is canonically isomorphic to $M\otimes {\mathcal{O}}_P(n)$, and $s_D$ is identified under this isomorphism with the section given by (**).

Corollary (2.9).

The subfunctor ${\mathrm{Div}}_{P/S}^n$ of ${\mathrm{Div}}_{P/S}^{+}$ corresponding to positive relative divisors of multidegree $n$ is canonically representable by the projective fibration $\mathbb{P}(({⨂}_{i}Sy{m}^{n_i}(E_i){)}^{\vee })$. Consequently ${\mathrm{Div}}_{P/S}^{+}$ is canonically representable by the sum-prescheme of the $\mathbb{P}(({⨂}_{i}Sy{m}^{n_i}(E_i){)}^{\vee })$ for $n\in {\mathbb{N}}^I$.

Representability via a sheaf $Q$ on the base

Remark (2.10).

The simplicity of the determination of ${\mathrm{Div}}_{P/S}^{+}$ above stems from the simplicity of the structure of $\mathrm{Pic}(P/S)$ — indeed from the "discrete" character of the Picard prescheme ${\mathrm{Pic}}_{P/S}$. We can abstract from the foregoing reasoning to a relative representability statement (relative to Pic).

To make this precise, take a proper flat morphism $f:X\to S$ of finite presentation and an invertible module $L$ over $X$. We propose to determine the subset ${\mathrm{Div}}^L(X/S)$ of ${\mathrm{Div}}^{+}(X/S)$ formed by those positive relative divisors $D$ such that ${\mathcal{O}}_X(D)$ is isomorphic to a module of the form $L{\otimes }_{{\mathcal{O}}_X}f\ast (M)$ for some invertible module $M$ over $S$ (depending on $D$). We assume that

$$(\ast \ast \ast ){\mathcal{O}}_S\stackrel{\sim }{\to }{f}_{\ast }({\mathcal{O}}_X),$$

which implies that the above $M$ is determined up to a unique isomorphism by $D$, namely

$$M=f_{\ast }(L^{-1}\otimes {\mathcal{O}}_X(D)).$$

To give $D$ corresponding to a given $L$ is then the same as to give a transversally regular section $\varphi$ of $L\otimes f\ast (M)$. By (III, §7), there exists a finitely presented module $Q$ over $S$ whose formation commutes with every base change, together with an isomorphism of functors in the quasi-coherent module $G$ over $S$,

  f_*(L ⊗ G) ≅ Hom(Q, G).

(In (III, §7) one supposes $S$ locally noetherian; this hypothesis is removed by the standard brief procedure, taking into account the base-change commutation of $Q$.) In particular, to give $\varphi$ is equivalent to giving a homomorphism $\Psi :Q\to M$. A necessary condition for $\varphi$ to be transversally regular — sufficient if the fibres of $X$ are integral — is that $\varphi$ should be $\ne 0$ fibre by fibre. In terms of $\Psi$, this means that $\Psi$ is surjective fibre by fibre, hence that $\Psi$ corresponds to a section of the projective fibration $\mathbb{P}(Q)$ over $S$.

We obtain:

Proposition (2.11).

In the above notations, ${\mathrm{Div}}^L(X/S)$ is in canonical bijection with the set of sections of $\mathbb{P}(Q)$ over $S$ — given by the quotient module $M$ of $Q$ — such that the section $\varphi$ of $L\otimes f\ast (M)$ defined by $\Psi :Q\to M$ is transversally regular.

Suppose now that the hypothesis (***) continues to hold after every base change; equivalently, by (III, §7), that

  k(s) ⥲ H^0(X_s, 𝒪_{X_s})  for every s ∈ S.

Then (2.11) applies equally well to every $X_{S^1}/S^1$ after an arbitrary base change $S^1\to S$. We obtain:

Theorem (2.12).

Let $f:X\to S$ be a proper morphism of finite presentation satisfying the above condition (***) universally, and let $L$ be an invertible module over $X$. Consider the subfunctor ${\mathrm{Div}}_{X/S}^L$ of ${\mathrm{Div}}_{X/S}^{+}$ defined above in terms of $L$. There exists a finitely presented module $Q$ over $S$ such that this subfunctor is representable by a retrocompact open sub-prescheme of the projective fibration $\mathbb{P}(Q)$. If, in addition, the fibres of $X/S$ are geometrically integral, then ${\mathrm{Div}}_{X/S}^L$ is representable by $\mathbb{P}(Q)$ itself.

The last assertion follows immediately from the construction above. For the general case we have already observed that there is a monomorphism ${\mathrm{Div}}_{X/S}^L\to \mathbb{P}(Q)$, and we are reduced to showing that this is a retrocompact open immersion. So we take a section of $\mathbb{P}(Q)$ over $S$ — i.e. an invertible quotient module $M$ of $Q$, giving a section $\varphi$ of $L\otimes f\ast (M)$ non-vanishing on every fibre — and we must show that the subfunctor of the final functor $S^{\prime }\mapsto Sc{h}_{/S}$ consisting of those $S^{\prime }\to S$ for which ${\varphi }_{S^{\prime }}$ is transversally regular is representable by a retrocompact open subset of $S$.

But this is now standard: representability by an open subset follows from the fact that $f$ is proper and that transverse regularity is an open condition (see (IV, §11)⁹); retrocompactness follows immediately by reduction to the noetherian case.

V.6.3. Linear systems of divisors and morphisms into projective fibrations

Fixed points of a family of divisors

(3.1). Let $D$ be a family of positive divisors over $X/S$ parametrized by $T$. A point $x\in X$ is called a fixed point of this family if $p{r}_1^{-1}(x)\subset D$ set-theoretically, so that the set of non-fixed points is the complement of $p{r}_1(X{\times }_{S}T-D)$. Consequently, if $T\to S$ is universally open (e.g. flat locally of finite presentation), the set of fixed points is closed. We say that the family of divisors is without fixed points if the set $Z$ of fixed points is empty.

If $Z$ is closed, then $X-Z$ is the largest open subset of $X$ such that the induced family of divisors of $(X-Z)/S$ parametrized by $T$ is without fixed points. If the family $D$ is without fixed points and $T$ is flat and locally of finite presentation over $S$ with geometrically irreducible fibres satisfying (S_1), then $D$ is also a relative divisor with respect to $X$ (for $p{r}_1:X{\times }_{S}T\to X$). Indeed, $D$ is defined locally at a point $z\in suppD$ by one equation $\varphi =0$, and the equation ${\varphi }_x$ induced on the fibre $T\otimes k(x)$ is non-nilpotent at $z$ (otherwise, $p{r}_1^{-1}(x)\cap D=V({\varphi }_x)$ would contain a neighbourhood of $z$ in $T\otimes k(x)$, hence the whole of $T\otimes k(x)$ since this fibre is irreducible — meaning $x$ would be a fixed point, which it is not). Since $T\otimes k(x)$ is irreducible and satisfies (S_1), it follows that ${\varphi }_x$ is ${\mathcal{O}}_{T\otimes k(x)}$-regular at $z$. We obtain therefore a family of divisors of $T/S$ parametrized by $X$, i.e. a morphism

$$X⟶{\mathrm{Div}}_{T/S}.$$

In the general case where the family of divisors of $X/S$ may have fixed points, we obtain a family of divisors of $T/S$ parametrized by $X-Z$,¹⁰ i.e. a morphism $X-Z\to {\mathrm{Div}}_{T/S}$, by replacing $X$ by $X-Z$ in the preceding construction. The argument shows in fact that $X-Z$ is exactly the greatest open subset $U$ of $X$ such that $D|U{\times }_{S}T$ is a relative divisor of $U{\times }_{S}T$ with respect to $U$ — equivalently, such that its "symmetric image" ᵗD is a family of divisors of $T/S$ parametrized by $U/S$.

We remark that if $X$ and $T$ are both flat locally of finite presentation over $S$, with geometrically irreducible fibres satisfying (S_1), then the symmetry $D\mapsto {}^{t}D$ gives a one-to-one correspondence between families of divisors of $X/S$ parametrized by $T$ without fixed points and families of divisors of $T/S$ parametrized by $X$ without fixed points.

To remove the hypothesis on the fibres of $X/S$ and $T/S$, it is convenient to replace "fixed points" by fixed points in the extended sense: by such a fixed point of $D$ we mean an $x\in X$ such that $D$ is not a relative divisor with respect to $X$ at all the points of $p{r}_1^{-1}(x)$. If $W$ is the open subset of $X{\times }_{S}T$ where $D$ is a relative divisor with respect to $X$, then the set of fixed points in the extended sense of $D$ equals $p{r}_1(X{\times }_{S}T-W)$; since $T\to S$ is proper, this is a closed subset $Z^{\prime }$ of $X$. In every case we obtain a family of divisors of $T/S$ parametrized by $X-Z^{\prime }$. The assumption that the fibres of $T/S$ are (S_1) and geometrically irreducible precisely ensures that $Z=Z^{\prime }$ (fixed points in the strict sense coincide with fixed points in the extended sense).

Geometric interpretation. Suppose for simplicity that $S=\mathrm{Spec}(k)$, with $k$ algebraically closed (which is permissible, after a base change, when $T/S$ is flat and of finite presentation). To say that $x\in X(k)$ is a fixed point (resp. fixed point in the extended sense) means that $x\in supp{D}_t$ for every $t\in T(k)$ (resp. that there exists a prime cycle $T^{\prime }$ associated to $T$ such that $x\in supp{D}_t$ for every $t\in T^{\prime }(k)$).

An omission. The formation of the set $Z$ of fixed points is compatible with base change in $S$; on the other hand, $X-Z$ (when open, e.g. for $T/S$ flat locally of finite presentation) is universally schematically dense in $X$ relative to $S$. This last fact follows from (IV, §11)¹¹ and from the fact that, for every $s\in S$, $Z_s$ contains no point of $X_s$ associated to ${\mathcal{O}}_{X_s}$ (the support of a divisor on $X_s$ contains no such point).

Linear systems and morphisms into projective fibrations

When $T$ is a projective fibration $Q=\mathbb{P}(F)$, the functor ${\mathrm{Div}}_{Q/S}^{+}$ is representable by the sum-prescheme of the $\mathbb{P}(Sy{m}^n(F^{\vee }))$ (which we denote $\mathbb{P}(n)$); we find then a morphism

  X − Z ⟶ ⨆_n ℙ(n).

We say that the family of divisors $D$ is of degree $n$ if the preceding morphism factors through $\mathbb{P}(n)$. If $X\ne \varnothing$ (so $X-Z\ne \varnothing$), the integer $n$ is well defined by $D$. To define this notion of degree, we strictly need only the canonical monomorphisms

$$\mathbb{P}(n)⟶{\mathrm{Div}}_{P^{\vee }/S}.$$

(Note. We set $P=\mathbb{P}(F^{\vee })$, so $Q=\mathbb{P}(F)=P^{\vee }$ with the notations of V.6.2.)

A linear system of divisors over $X/S$ parametrized by the projective fibration $Q=P^{\vee }$ is a family of divisors over $X/S$ parametrized by $Q$ which is of degree one, i.e. defining $f:X-Z\to P$. To such a linear system, when the fibres of $P^{\vee }$ are non-empty,¹² is associated a rational map (better: a "pseudo-morphism relative to $S$") $X⇢P$.

By the very construction, $D|(X-Z){\times }_S{P}^{\vee }$ is the inverse image, under $f\times i{d}_{P^{\vee }}$, of the canonical incidence divisor $H$ over $P{\times }_S{P}^{\vee }$. Hence the knowledge of $f:X-Z\to P$ allows us to reconstruct, at least, the family of divisors of $(X-Z)/S$ induced by $D$. So if the family is without fixed points it is determined by the associated morphism $f:X\to P$. We obtain a one-to-one correspondence between linear systems without fixed points over $X/S$ parametrized by $P^{\vee }$ and morphisms $f:X\to P$ such that $(f\times id{)}^{-1}(H)$ is a relative divisor of $X{\times }_S{P}^{\vee }$ with respect to $P^{\vee }$. This condition can be checked fibre by fibre:

Proposition (3.1).

We have a one-to-one correspondence between linear systems without fixed points of divisors over $X/S$ parametrized by $P^{\vee }$ and morphisms $f:X\to P$ having the following property: for every $s\in S$, denoting by $\overline{k(s)}$ an algebraic closure of $k(s)$, and for every prime cycle $X^{\prime }$ associated to $X_{\overline{k(s)}}$, $f(X^{\prime })$ is not contained in any hyperplane of $P_{\overline{k(s)}}$. (If $X$ has geometrically integral fibres, this can be stated simply by saying that $f(X_{\overline{k(s)}})$ is not contained in any hyperplane of $P_{\overline{k(s)}}$.)

In general ($Z\ne \varnothing$) we can no longer assert that the knowledge of $f$ determines the family of divisors. The most trivial case is $P^{\vee }=S$ of relative dimension zero: a linear system of divisors of $X/S$ parametrized by $S$ is just a relative Cartier divisor $D$ over $X$ relative to $S$, the associated morphism is the projection $X-D\to S$, and the knowledge of this morphism (including its domain of definition) does not determine $D$.

To eliminate such unpleasant phenomena, we limit ourselves to linear systems without fixed components.

If $S=\mathrm{Spec}(k)$, then given a family (not necessarily linear) of divisors of $X/S$ parametrized by $T$, a fixed component of the family is any irreducible component of codimension one of the set $Z$ of fixed points; the family is without fixed components if it has no fixed component, i.e. $codim(Z,X)\ge 2$. This terminology extends immediately to arbitrary $S$ by considering fibres. The property of having no fixed component is stable under base change.

Proposition (3.2).

Suppose $X\to S$ is flat and locally of finite presentation, with fibres satisfying (S_2). Let $D$ be a linear system of divisors without fixed components over $X/S$ parametrized by $P^{\vee }$. Then $D$ is uniquely determined by the corresponding morphism $f:X-Z\to P$ ($Z$ = set of fixed points), and even by the equivalence class of $f$ as a pseudo-morphism relative to $S$; the set $X-Z$ is the domain of definition of that class.¹³

We must prove that if $D^{\prime }$ is another linear system of divisors without fixed components parametrized by $P^{\vee }$, defining $f^{\prime }:X-Z^{\prime }\to P$, and if $f$ and $f^{\prime }$ agree on an open subset $U\subset (X-Z)\cap (X-Z^{\prime })$ schematically dense relative to $S$, then $D=D^{\prime }$. Since $P$ is separated over $S$, we may take $U=(X-Z)\cap (X-Z^{\prime })=X-Z^{\prime \prime }$, where $Z^{\prime \prime }=Z\cup Z^{\prime }$. Since Z'' is of codimension $\ge 2$ on each fibre and $X$ has (S_2) fibres, for every $x\in Z^{\prime \prime }$ the fibre $X_s$ has depth $\ge 2$ at $x$. We may conclude (using flatness of $X$ over $S$) that every divisor over $X$ (not necessarily transversal to the fibres) is determined by its restriction to $X-Z^{\prime \prime }$.

Let $\mathcal{J}$ be the ideal sheaf defining $D$; it suffices to show that $\mathcal{J}\to i_{\ast }(\mathcal{J}|X-Z^{\prime \prime })$ is an isomorphism (where $i:X-Z^{\prime \prime }\hookrightarrow X$ is the canonical immersion), for then the homomorphism $\mathcal{J}\to {\mathcal{O}}_X$ is recovered by applying $i_{\ast }$ to $\mathcal{J}|X-Z^{\prime \prime }\to {\mathcal{O}}_X|X-Z^{\prime \prime }$. Since $\mathcal{J}$ is invertible it is flat over $S$, and for $x\in Z^{\prime \prime }$, $pro{f}_x{\mathcal{J}}_{(s)}\ge 2$. It is therefore enough to prove:

Lemma (3.3).

Let $X\to S$ be of finite presentation, $\mathcal{F}$ a module of finite presentation over $X$, flat relative to $S$, and $T$ a closed subset of $X$. Assume that for every $x\in X$ over $s\in S$ we have $pro{f}_x{\mathcal{F}}_s\ge 1$ (resp. $\ge 2$). Then the canonical homomorphism $\mathcal{F}\to i_{\ast }(\mathcal{F}|X-T)$ is injective (resp. bijective), where $i:X-T\hookrightarrow X$ is the canonical immersion.

Indeed, we may suppose $S$ and $X$ affine, and by the brief procedure reduce to $S$ noetherian. Then the hypothesis implies, by (IV, §6), that $pro{f}_x\mathcal{F}\ge pro{f}_x{\mathcal{F}}_s$ for every $x\in X$ over $s\in S$, so $pro{f}_x\mathcal{F}\ge 1$ (resp. $\ge 2$) for $x\in T$. We conclude by (IV, §5).¹⁴

Grothendieck note. For best results, this lemma ought to appear in (IV, §11) ("elimination of the noetherian hypothesis…"); compare (IV, 11.3).¹⁵

It finally remains to verify the last assertion of Proposition (3.2): that $X-Z$ is exactly the domain of definition of the rational map relative to $S$ defined by $f$. Let $U\supset X-Z$ be its domain of definition. By Proposition (3.1), $U\to P$ is associated to a linear system of divisors D'' over $U/S$ parametrized by $P^{\vee }$, and $D^{\prime \prime }|(X-Z){\times }_S{P}^{\vee }=D|(X-Z){\times }_S{P}^{\vee }$. Applying the uniqueness result to D'' and $D|U{\times }_S{P}^{\vee }$, the two are equal, so $D|U{\times }_S{P}^{\vee }$ has no fixed points: $Z\cap U=\varnothing$, hence $U=X-Z$. q.e.d.

Grothendieck note. I repent of having stated the proposition in a muddled form, half-way between the classical hypotheses and the natural ones, and without giving the converse.¹⁶ So I propose to announce instead:

Proposition (3.4).

Let $X\to S$ be flat locally of finite presentation, $Q=P^{\vee }=\mathbb{P}(E^{\vee })$ a projective fibration over $S$ defined by a locally free module of finite type $F=E^{\vee }$. Consider the set $\Phi$ of linear systems $D$ of divisors over $X$ parametrized by $Q$ such that the set $Z$ of fixed points satisfies: $z\in Z⟹pro{f}_z{\mathcal{O}}_{X_s}\ge 2$ (where $s$ is the image of $z$ in $S$). Consider also the set $\mathfrak{U}$ of pseudo-morphisms $f$ relative to $S$ of $X$ into $P$ such that the domain of definition $U=U(f)$ satisfies $z\in X-U⟹pro{f}_z{\mathcal{O}}_{X_s}\ge 2$, and such that $f_U=f|U:U\to P$ satisfies the non-degeneracy of (3.1). Consider the natural map $D\mapsto f_D$ of $\Phi$ into $\mathfrak{U}$. Then:

(a) This map is injective, and for $D\in \Phi$ the set of fixed points $Z$ is just the complement $X-U$ of the domain of definition of $f_D$.

(b) Let $f\in \mathfrak{U}$, and let $U$ be an open subset over which $f$ is defined and such that $z\in X-U⟹pro{f}_z{\mathcal{O}}_{X_s,z}\ge 2$ (for instance $U=U(f)$). In order that $f$ give rise to some $D\in \Phi$, it is necessary and sufficient that, setting $L_U=f_U^{\ast }({\mathcal{O}}_P(1))$ (with $f_U:U\to P$ the morphism induced by $f$), the module $i_{\ast }(L_U)$ over $X$ is invertible (where $i:U\hookrightarrow X$ is the canonical immersion).

If the fibres of $X$ over $S$ satisfy (S_2) — for instance if they are normal, or even just geometrically normal — then the depth condition on a closed subset $Z$ of $X$ in the proposition means simply that for every $s\in S$, $Z_s$ is of codimension $\ge 2$ in $X_s$; $\Phi$ is therefore exactly the set of linear systems without fixed components. Moreover, if $S=\mathrm{Spec}(k)$ and $X$ is normal, then for every rational map $f:X⇢P$ the domain of definition $U(f)$ satisfies $codim(X-U(f),X)\ge 2$ (II.7), so $\mathfrak{U}$ is the set of all rational maps $X⇢P$.

The proof of (a) has already been given. For (b), we note that the formation of $i_{\ast }(L_U)$ commutes with every flat extension $S^{\prime }\to S$ (at least if $U\to X$ is quasi-compact, the case to which we reduce without difficulty), so the condition in question is invariant under faithfully flat quasi-compact base change. Taking $S^{\prime }=P^{\vee }$ and noting that the hypothesis that $i_{\ast }^{\prime }(L_U^{\prime })$ is invertible does not change if we replace $L_U^{\prime }$ by $L_U^{\prime }{\otimes }_{S^{\prime }}{M}^{\prime }$ for any invertible $M^{\prime }$ over $S^{\prime }$, we have

  i'_*(L'_U ⊗_{S'} M') ≅ i'_*(L'_U) ⊗_{S'} M'.

Taking $M^{\prime }={\mathcal{O}}_{P^{\vee }}(1)$, the condition becomes that $i_{\ast }^{\prime }(N^{\prime })$ is invertible, where

  N' = (f_U × id_{P^∨})^*(𝒪_{P ×_S P^∨}(1, 1)).

But $\mathcal{O}(1,1)$ is precisely the invertible module defined by the canonical (incidence) divisor $H$ of $P{\times }_S{P}^{\vee }$, so that $N^{\prime }$ is the invertible module defined by $D^{\prime }=(f_U\times i{d}_{P^{\vee }}{)}^{-1}(H)$. If $f$ gives rise to a $D\in \Phi$, then $D^{\prime }=D|U{\times }_S{P}^{\vee }$ and $N^{\prime }=N|U{\times }_S{P}^{\vee }$, where $N$ is the invertible module over $X{\times }_S{P}^{\vee }$ defined by $D$. It follows from Lemma (3.3), applied to $P{\times }_S{P}^{\vee }\to P$, that $i_{\ast }(N^{\prime })\cong N$; hence $i_{\ast }^{\prime }(N^{\prime })$ is invertible.

Conversely, if this condition is satisfied, we show that $f$ gives rise to a $D\in \Phi$, which evidently amounts to saying that $D^{\prime }$ extends to a relative divisor of $X{\times }_S{P}^{\vee }$ with respect to $P^{\vee }$. It suffices to show that it extends to a divisor $D$ over $X{\times }_S{P}^{\vee }$ (it will then automatically be a relative divisor with base $P^{\vee }$, since $U$ contains elements associated to ${\mathcal{O}}_{X_s}$ for every $s\in S$, and that property is stable under base change — in particular under $S^{\prime }=P^{\vee }\to S$). But by Lemma (3.3) again, $D^{\prime }$ extends to a divisor $D$ if and only if $D^{\prime }$ extends to an invertible module, i.e. iff $i_{\ast }^{\prime }(N^{\prime })$ is invertible. We have therefore the necessary and sufficient condition.

Grothendieck note. The end of the proof should be edited to express the necessary and sufficient condition once only (rather than twice as I did), and to begin by isolating the following corollary of Lemma (3.3).

Corollary (3.5).

Suppose $g:X\to S$ is flat and locally of finite presentation. Let $T$ be a closed subset of $X$ such that $x\in T⟹pro{f}_x{\mathcal{O}}_{X_s}\ge 2$ (where $s=g(x)$), and set $U=X-T$, $i:U\hookrightarrow X$. For every locally free module $L$ of finite type over $X$, let $L^{\prime }=L|U$. Then:

(a) The functor $L\mapsto L^{\prime }$ is fully faithful, and for every $L$ the canonical homomorphism $L\to i_{\ast }(L^{\prime })$ is an isomorphism. For $L$ to be of rank $n$, it is necessary and sufficient that $L^{\prime }$ should be.

(b) A locally free module $L^{\prime }$ over $U$ is isomorphic to the restriction of a locally free module $L$ over $X$ if and only if $i_{\ast }(L^{\prime })$ is locally free.

(c) Suppose $L^{\prime }$ is an invertible module associated to a divisor $D^{\prime }$ over $U$. Then the condition of (b) is also necessary and sufficient in order that $D^{\prime }$ should be the restriction of a divisor $D$ over $X$ (which is then unique, and equals the scheme-theoretic closure of $D^{\prime }$ in $X$). For $D$ to be a relative divisor with respect to $S$, it is necessary and sufficient that $D^{\prime }$ should be.

We simply use the fact that every such $L$ satisfies the hypothesis on $\mathcal{F}$ in Lemma (3.3).

Corollary (3.6).

Suppose the local rings of $X$ are factorial (for example, $X$ regular). Then the map $\Phi \to \mathfrak{U}$ of (3.4) is bijective. In particular, if $X$ is a regular prescheme locally of finite type over a field $k$, and $P$ is a projective fibration over $k$, then there is a one-to-one correspondence between the set $\Phi$ of linear systems of divisors without fixed components over $X$ parametrized by $P^{\vee }$, and the set $\mathfrak{U}$ of rational maps $X⇢P$ whose image (over $\overline{k}$) does not factor through any hyperplane of $P_{\overline{k}}$.

Indeed, since the local rings of $X$ are factorial, every invertible module over $U$ extends to an invertible module over $X$, so the condition of (3.4)(b) is automatically satisfied. (By Auslander-Buchsbaum, a regular local ring is factorial.)

V.6.4. Linear systems of divisors and invertible modules

Using the results of V.6.1, we shall give a complete description of linear systems over $X$ in terms of invertible sheaves over $X$.

We may suppose that $P^{\vee }\to S$ is surjective; then so is $X{\times }_S{P}^{\vee }\to P^{\vee }$. According to the generalities of (V, 6.2) above, to give a divisor $D$ over $X{\times }_S{P}^{\vee }$ is to give an invertible module $N$ over $X{\times }_S{P}^{\vee }$ together with a regular section $\varphi$ of $N$.¹⁷ The assumption that $D$ is a linear system of divisors over $X$ parametrized by $P^{\vee }$ can be expressed by the two conditions

1. (regularity fibre by fibre) the sections ${\varphi }_t$ ($t\in P^{\vee }$) induced by $\varphi$ on the fibres of $X{\times }_S{P}^{\vee }$ over $P^{\vee }$ are regular (which entails that $\varphi$ is regular); and
2. (degree 1) the modules $N_x$ ($x\in X$) induced by $N$ on the fibres of $X{\times }_S{P}^{\vee }$ over $X$ are of degree one.

By V.6.1, to give an $N$ invertible over the projective fibration $X{\times }_S{P}^{\vee }$ over $X$ satisfying condition 2 is equivalent to giving an invertible module $L$ over $X$; $N$ is then determined as a function of $L$ by

  N = L ⊗_{𝒪_S} 𝒪_{P^∨}(1),

and $L$ is determined in terms of $N$ by

$$L\cong p{r}_{1,\ast }(N(-1))$$

(where $(-1)$ denotes tensoring with ${\mathcal{O}}_{P^{\vee }}(-1)$ over ${\mathcal{O}}_S$). To give $\varphi$ is then to give a section of $L\otimes {\mathcal{O}}_P(1)$, i.e. a section of

  pr_{1, *}(L ⊗_{𝒪_X} 𝒪_{X ×_S P^∨}(1)) = L ⊗_{𝒪_S} pr_{1, *}(𝒪_{X ×_S P^∨}(1)).

By (III, §2), $p{r}_{1,\ast }({\mathcal{O}}_{X{\times }_S{P}^{\vee }}(1))=E_X^{\vee }$, so that to give $\varphi$ is equivalent to giving a morphism $g\ast (E)\to L$ or, what is the same, a morphism $u:E\to g_{\ast }(L)$, where $g:X\to S$ is the canonical projection.

It remains to express condition 1 in terms of $u$. Since the construction commutes with base change, it suffices to express this condition fibre by fibre, taking into account that the points of $P$ with values in an extension $k$ of $k(s)$ correspond exactly to lines in $E(s){\otimes }_{k(s)}k$. This condition is: for every $t\in E(s)$, the corresponding section $u(s)(t)$ of $L_s$ over $X_s$ is regular, and the analogous condition holds after every extension of the base field. As usual, it suffices to test this over an algebraic closure of $k(s)$.

To summarize:

Proposition (4.1).

Let $g:X\to S$ be a flat morphism locally of finite presentation, $P=\mathbb{P}(E)$ a projective fibration over $S$ defined by $E$ locally free of finite type, everywhere $\ne 0$ (so that $P$ has non-empty fibres), and $P^{\vee }=\mathbb{P}(E^{\vee })$. Then there is a bijection between the set of linear systems of divisors over $X/S$ parametrized by $P^{\vee }$ and the set of pairs $(L,u)$ (up to isomorphism), where $L$ is an invertible module over $X$ and $u:E\to g_{\ast }(L)$ is a homomorphism such that for every $s\in S$ and every point $t$ of $E(s){\otimes }_{k(s)}\overline{k(s)}$ ($\overline{k(s)}$ an algebraic closure of $k(s)$), the corresponding section $u(t)$ of $L_s\otimes \overline{k(s)}$ over $X_s\otimes \overline{k(s)}$ is regular.

If the fibres of $X$ are geometrically integral, the condition on $u$ simplifies: for every $s\in S$, $u(s):E(s)\to H^0(X_s,L_s)$ is injective. For convenience of reference, we recall the construction of the divisor $D$ in terms of $(L,u)$: it is the divisor of the evident section $\varphi$ of $L{\otimes }_{{\mathcal{O}}_S}{\mathcal{O}}_{P^{\vee }}(1)$ defined by $u$.

Corollary (4.2).

Assume $g:X\to S$ is proper, flat, and of finite presentation with geometrically integral fibres. Let $L$ be an invertible module over $X$ and $P=\mathbb{P}(E)$ a projective fibration over $S$ as in (4.1). There exists a finitely presented module $Q$ over $S$ and an isomorphism of functors of the quasi-coherent ${\mathcal{O}}_S$-module $F$:

  Hom(Q, F) ⥲ g_*(L ⊗_{𝒪_S} F).

With this in hand, the linear systems of divisors on $X$ parametrized by $P^{\vee }$ and associated to $L$ in the sense of (4.1) correspond bijectively to the surjective homomorphisms $Q\to E^{\vee }$ modulo multiplication by a section of ${\mathcal{O}}_S^{\times }$.

The existence of $Q$ reduces by the brief procedure to the case $S$ noetherian, where it is (III, 7.7.6) (the hypothesis on the fibres of $X$ being unnecessary). Since $E$ is locally free of finite type, to give a homomorphism $E\to g_{\ast }(L)$ is the same as to give a section of $g_{\ast }(L)\otimes E^{\vee }$, i.e. a homomorphism $Q\to E^{\vee }$. It remains to express the condition of (4.1) in this form. By hypothesis on the fibres of $X/S$, this reduces to fibre-by-fibre injectivity of

  E(s) ⟶ H^0(X_s, L_s) ≅ Hom_{k(s)}(Q(s), k(s)),

i.e. surjectivity of $Q(s)\to E^{\vee }(s)$, which by Nakayama means surjectivity of $Q\to E^{\vee }$. The "modulo sections of ${\mathcal{O}}_S^{\times }$" becomes "modulo isomorphisms" in (4.1).

We may interpret (4.2) in another way using the fact that $\mathbb{P}(Q)$ represents the subfunctor of ${\mathrm{Div}}_{X/S}$ defined by $L$, by virtue of V.6.2. Consequently, a linear system parametrized by $P^{\vee }$ and associated to $L$ is interpreted as a morphism $P^{\vee }=\mathbb{P}(E)\to \mathbb{P}(Q)$. The "linear" character of the family of divisors so defined is captured by saying that this morphism is linear, i.e. is defined by a surjective morphism $Q\to E^{\vee }$. In this case $P^{\vee }\to {\mathrm{Div}}_{X/S}$ is also a monomorphism (since $P^{\vee }\to \mathbb{P}(Q)\to {\mathrm{Div}}_{X/S}$ is such); this is a more general fact, cf. the corollary below.

We agree that two linear families of divisors of $X/S$ parametrized by projective fibrations $P^{\vee }$, $(P^{\vee }{)}^{\prime }$ are isomorphic if they are transformed into each other by an $S$-isomorphism $P^{\vee }\stackrel{\sim }{\to }(P^{\vee }{)}^{\prime }$ (which is unique, by the monomorphism property). We may then express (4.2) by saying that the set of classes (up to isomorphism) of linear systems of divisors over $X$ associated to $L$ is in canonical bijection with the set $Grass(Q)(S)$, and this correspondence is compatible with base change. The functor

  S' ⟼ \{classes (mod isomorphism) of linear systems of divisors of X_{S'}/S' associated to L_{S'}\}

is therefore representable by the $S$-prescheme $Grass(Q)$.

Grothendieck note. (Marginal remarks here are hard to read.)¹⁸

We should also make explicit in (4.1) that $L|X-Z$ is canonically isomorphic to $f\ast ({\mathcal{O}}_P(1))$ (with the notations of V.6.3). So for a $D\in \Phi$ in the sense of (3.4), $L|X-Z$ is exactly the canonical and unique extension of $f\ast ({\mathcal{O}}_P(1))$ to an invertible sheaf over $X$.

Monomorphism into ${\mathrm{Div}}_{X/S}$

Proposition (4.3).

Let $D$ be a linear system of divisors over $X/S$ parametrized by $P^{\vee }$, with $g:X\to S$ flat of finite presentation.

(a) Suppose $g$ is of finite presentation, and that for every $s\in S$, denoting by $\overline{k(s)}$ an algebraic closure of $k(s)$, there exists a prime cycle $T$ associated to $X_{\overline{k(s)}}$ such that $\overline{k(s)}\stackrel{\sim }{\to }{H}^0(T,{\mathcal{O}}_T)$ (a condition automatically satisfied if $g$ is proper and surjective). Then the morphism $D:P^{\vee }\to {\mathrm{Div}}_{X/S}$ is a monomorphism.

(b) Consider the map $u\mapsto D\circ u$ from ${\mathrm{Aut}}_S(P^{\vee })$ into the set of families of divisors over $X/S$ parametrized by $P^{\vee }$. If $g$ is surjective, this map is injective; in particular, $D=D\circ u$ implies $u=i{d}_{P^{\vee }}$. More generally, the morphism of functors Aut_S(P^∨) → \{linear systems of divisors of X/S parametrized by P^∨\} is a monomorphism.

We note that under the hypotheses of (a), (b) is a trivial consequence of (a); on the other hand, (b) holds under less restrictive assumptions than (a). The hypothesis in (a) is genuinely needed: for example, take $S=\mathrm{Spec}(k)$ and $X$ an open subset of ${\mathbb{P}}_k^1$ not containing two distinct points $a,b\in {\mathbb{P}}_k^1(k)$; then $a$ and $b$ define the same divisor of $X$ (the zero divisor!) without being identical.

Suppose first that $S=\mathrm{Spec}(k)$ with $k$ algebraically closed (a reduction we may make by descent). Let $T$ be as in (a), equipped with the induced reduced structure; we have a morphism

$${\mathrm{Div}}_{X/k}⟶{\mathrm{Div}}_{T/k}$$

("induced divisor"), and it suffices to show that the composition $P^{\vee }\to {\mathrm{Div}}_{T/k}$ is a monomorphism. The latter is again a linear system of divisors, so we are reduced to the case $X=T$, hence to the case where $H^0(X,{\mathcal{O}}_X)\stackrel{\sim }{\to }{\mathcal{O}}_S$. Then for every $S^{\prime }$ over $k$,

$$g_{S^{\prime },\ast }({\mathcal{O}}_{X_{S^{\prime }}})\stackrel{\sim }{\to }{\mathcal{O}}_{S^{\prime }}.$$

Now if $L$ over $X$ and $u:E\to g_{\ast }(L)$ are as in (4.1), and if two sections $\varphi$, $\psi$ of $E_{S^{\prime }}$ everywhere non-zero are such that $u(\varphi )$ and $u(\psi )$ have the same divisor as sections of $L_{S^{\prime }}$ over $X_{S^{\prime }}$, then $u(\psi )$ is deduced from $u(\varphi )$ by multiplication by an invertible section of ${\mathcal{O}}_{X_{S^{\prime }}}$; from $(g_{S^{\prime }}{)}_{\ast }({\mathcal{O}}_{X_{S^{\prime }}})\stackrel{\sim }{\to }{\mathcal{O}}_{S^{\prime }}$, this multiplier is an invertible section of ${\mathcal{O}}_{S^{\prime }}$. So $\psi =c\cdot \varphi$ for $c\in {\mathcal{O}}_{S^{\prime }}^{\times }$, meaning $\varphi$ and $\psi$ define the same point of $P^{\vee }$ with values in $S^{\prime }$. Since every point of $P^{\vee }$ with values in $S^{\prime }$ is defined locally over $S^{\prime }$ by a section $\varphi$ of $E_{S^{\prime }}$ not vanishing anywhere (cf. Chapter I), (a) follows.

To prove (b) we record:

Lemma (4.4).

Let $D$ be a non-empty linear system of divisors over $X$ locally of finite type over an algebraically closed field $k$, parametrized by $P^{\vee }=\mathbb{P}(E)$, and let $f:X-Z\to P$ be the corresponding morphism ($Z$ = base locus, i.e. set of fixed points). If $r=ran{k}_{k}E>0$, there exist $r+1$ points $x_i$ ($1\le i\le r+1$) of $X(k)-Z(k)$ such that the $f(x_i)$ form a "projective base" of $P$: for every subset $J\subset [1,r+1]$ of cardinality $r$, the $f(x_j)$ ($j\in J$) are not contained in any hyperplane of $P$.

We may suppose $Z=\varnothing$. Since (by (4.1)) $f(X)$ is not contained in any hyperplane of $P$, we obtain at once the existence of $r$ points $x_i$ ($1\le i\le r$) such that the $f(x_i)$ are projectively independent in $P$, i.e. defined by linearly independent forms over $E$. It remains to find $x_{r+1}=x\in X(k)$ such that $f(x)$ lies in none of the $r$ hyperplanes $H_i$ defined by the $(r-1)$-tuples among the $f(x_j)$. Suppose the contrary; then by Jacobson-type sorites,

$$f(X)\subset \bigcup _i{H}_i,$$

so for some irreducible component $X_0\subset X$, $f(X_0)\subset H_i$ for some $i$, contradicting (3.1) (or (4.1)). q.e.d.

To prove (b), we may suppose $Z=\varnothing$. By (3.1) (or (4.1)), we are reduced to showing that an automorphism $u$ of $P^{\vee }$ is determined by the composition of its contragredient $u^{\vee }$ (on $P$) with $f:X\to P$, and that the analogous statement holds after every base change $S\to \mathrm{Spec}(k)$ by an automorphism $u$ of $P_S^{\vee }$. But this follows immediately from Lemma (4.4) and from the determination of automorphisms of $\mathbb{P}(E^{\vee })=\mathbb{P}(E{)}^{\vee }$ in V.6.1: the effect of an automorphism of a projective fibration over $S$ (relative to a locally free module of finite type) is determined by its effect on a projective base in each fibre.

The general case: $S$ arbitrary

By base change over $S$, we are reduced in (a) to showing that any two sections of $P^{\vee }$ over $S$ defining the same divisor of $X$ are identical, and in (b) to showing that any two automorphisms of $P^{\vee }$ for which $D\circ u=D\circ v$ are identical. We may suppose $S$ affine; in (b), where we do not expressly suppose $g$ of finite presentation but $g$ surjective, we reduce immediately (using that $g$ is open) to the case where $X$ is also affine, hence of finite presentation over $S$. By the brief procedure we reduce to $S$ noetherian.