Éléments de Géométrie Algébrique — English

§V.5. Hyperplane sections and conic projections (formerly EGA IV §20) — part 1 of 2

This section was originally drafted as §20 of EGA IV, then re-allocated to EGA V (Chapter V §5) without ever being published in either place. It is the longest single section of the prenotes. Because of its bulk we divide our translation into two files: this part 1 covers the introductory plan and §§V.5.1-V.5.8 (preliminaries, the generic hyperplane section, sufficiently general hyperplane sections, Seidenberg-type theorems, connectedness of an arbitrary hyperplane section, applications to the construction of special hyperplane sections and multisections, and the dimension of the set of exceptional hyperplanes); part 2 (the companion file 04-ch5-05-hyperplane-sections-part-2.md) covers §§V.5.9-V.5.16 (change of projective embedding, pencils of hyperplane sections, Grassmannians, linear sections, M. Artin's theorem on elementary morphisms, conic projections, axiomatization, and translation into the language of linear systems).

The §V↔§IV correspondence is given in the front matter; we lead with the V numbering and attach (formerly IV, M) parenthetically at the first occurrence of each cross-reference into the old numbering.

Grothendieck note (placed at the head of the prenote as a summary). This formulation gives, pell-mell,¹ a detailed summary of the set of results that should appear in a final formulation. To arrive at the latter we need to reorganize thoroughly the present stage zero. The first step should probably be to make a new plan (in which without a doubt the present §§5.11, 5.12, 5.14, 5.15 will come much earlier). I have not even written §5.16, which should neither in principle cause any difficulty nor influence in any way the previous numbers, since what is involved is a simple matter of translation.²

You will notice the presence of a Proposition (5.10.3) which should appear in a previous paragraph.

I would like to tell you in this connection that I have several other results, quite diverse but all dealing with birational mappings, that I would love to include somewhere. It seems to me that there is not enough material to make a paragraph on its own. Do you have a suggestion where to place them? I plan to send them to you soon, as well as §5.16 of the present notes.

In addition, the present §V.5 (Paragraph 20) will probably blow up into two paragraphs, one consisting of results of "elementary-geometry" type on Grassmannians. If need be, could one include there also (lacking a better place) the supplements that I told you about dealing with birational transformations?

Plan of §V.5

1. Preliminaries and notation (§V.5.1).

2. Generic hyperplane section: local properties (§V.5.2).

3. Generic hyperplane section: geometric irreducibility and connectedness (§V.5.3).

4. Variable hyperplane section: "sufficiently general" sections (§V.5.4).

5. Theorems of Seidenberg type (§V.5.5).

6. Connectedness of an arbitrary hyperplane section (§V.5.6).

7. Application to the construction of special hyperplane sections and multisections (§V.5.7).

8. Dimension of the set of exceptional hyperplanes (§V.5.8).

9. Change of projective embedding (§V.5.9).

10. Pencils of hyperplane sections and fibrations of blown-up varieties (§V.5.10).

11. Grassmannians (§V.5.11).

12. Generalization of the previously mentioned results to linear sections (§V.5.12).

13. Elementary morphisms and a theorem of M. Artin (§V.5.13).

14. Conic projections (§V.5.14).

15. Axiomatization of some of the previous results (§V.5.15).

16. Translation into the language of linear systems (§V.5.16).

Items 1-8 are treated in the present part 1; items 9-16 are treated in part 2.

V.5.1. Preliminaries and notation

Let $S$ be a prescheme, let $E$ be a locally free module of finite type over $S$, and let $E^{\vee }$ be its dual. We denote by $P=P(E)=\mathbb{P}(E)$ the projective fibration defined by $E$, and by $P^{\vee }=\mathbb{P}(E^{\vee })$ the projective fibration defined by $E^{\vee }$. We shall call $P^{\vee }$ the scheme of hyperplanes of $P$. This terminology can be justified as follows. Let $\xi$ be a section of $P^{\vee }$ over $S$, determined by an invertible quotient module $L$ of $E^{\vee }$. From it we obtain an invertible quotient module L_P of $(E^{\vee }{)}_P=(E_P{)}^{\vee }$; on the other hand, we have the invertible quotient module ${\mathcal{O}}_P(1)$ of E_P. Passing to duals, we may take $L_P^{-1}$ and ${\mathcal{O}}_P(-1)$ to be invertible submodules (locally direct factors) of E_P and of $(E_P{)}^{\vee }$ respectively, and the pairing $E_P\otimes E_P^{\vee }\to {\mathcal{O}}_P$ defines therefore a natural pairing

$$(\ast ){\mathcal{O}}_P(-1)\otimes L_P^{-1}⟶{\mathcal{O}}_P,$$

or equivalently the transposed homomorphism

  (**)                           𝒪_P ⟶ 𝒪_P(1) ⊗ L_P = L_P(1),

that is, a section of $L_P(1)$ canonically defined by $\xi$. The "divisor" of this section, i.e. the closed subscheme $H_{\xi }$ of $P$ defined by the image ideal of (*), is called the hyperplane in $P$ defined by the element $\xi \in P^{\vee }(S)$. We could describe it by noting that, locally over $S$, $\xi$ is given by a section $\varphi$ of $E$ such that $\varphi (s)\ne 0$ for all $s$ ($\varphi$ is determined by $\xi$ up to multiplication by an invertible section of ${\mathcal{O}}_S$); since $E=p_{\ast }({\mathcal{O}}_P(1))$ ($p:P\to S$ being the projection), $\varphi$ can be considered as a section of ${\mathcal{O}}_P(1)$, and the divisor of this section is nothing else but $H_{\xi }$.

Of course, if we consider $L^{-1}$ as an invertible submodule of $E$, locally a direct factor in $E$, then the correspondence between $\xi$ (that is, $L$, or $L^{-1}\subset E$) and $\varphi$ is obtained by taking for $\varphi$ a section of $L^{-1}$ which does not vanish at any point — i.e. by a trivialization of $L^{-1}$ (which exists in every case locally). Let us note that $H_{\xi }$ is nothing else but $\mathbb{P}(E/L^{-1})$ (canonical isomorphism); this is a third way of describing $H_{\xi }$. (N.B. $\mathbb{P}(E/L^{-1})$ is indeed canonically embedded in $P=\mathbb{P}(E)$, which has the advantage of proving in addition that $H_{\xi }$ is a projective fibration over $S$ and is a fortiori smooth over $S$. It would still be necessary to say in §17 of EGA IV that a projective fibration is smooth.) Without a doubt it would be better to begin with this description.

Remarks (5.1.1).

The formation of $H_{\xi }$ is compatible with base change; in other words one finds a homomorphism of functors $(Sch/S)\to (Sets)$,

$$P^{\vee }⟶\mathrm{Div}(P/S),$$

where the second term denotes the functor of "relative divisors" of $P/S$, whose value at an arbitrary $S$-prescheme $S^{\prime }$ is the set of closed subschemes of $P_{S^{\prime }}$ which are complete intersections, transversal to and of codimension 1 relative to $S^{\prime }$ (compare §V.19, formerly IV, 19).³

It is easy to show that this homomorphism of functors is a monomorphism — in other words, that $\xi$ is determined by $H_{\xi }$. (This last fact justifies the terminology "scheme of hyperplanes" used above.) We shall see that the functor $\mathrm{Div}(P/S)$ is representable by the prescheme (direct) sum of the $\mathbb{P}(Sy{m}^k(E^{\vee }))$, so that $P^{\vee }$ can be identified to an open and closed subscheme of $\mathrm{Div}(P/S)$. (Compare Mumford, Lectures on curves on an algebraic surface.) (N.B. To tell the truth, the determination of the relative divisors of $P/S$ could be done with the means available right now, using results of Chapter II, and could be added as an example to §IV.19.)

Let us now make the base change $S^{\prime }=P^{\vee }\to S$ and consider the diagonal section (or "generic section") of $P_{S^{\prime }}^{\vee }=\mathbb{P}(E_{S^{\prime }}^{\vee })$ over $S^{\prime }$: we find a closed subscheme $H$ of $P_{S^{\prime }}=P{\times }_S{P}^{\vee }$ — sometimes called the incidence scheme between $P$ and $P^{\vee }$ — defined by the image ideal of the canonical homomorphism

  𝒪_P(-1) ⊗_S 𝒪_{P^∨}(-1) ⟶ 𝒪_{P ×_S P^∨},

from which we see that it is a projective fibration over $P^{\vee }$; by symmetry it is also a projective fibration over $P$. Note that one recovers the "special" hyperplanes $H_{\xi }$ (for $\xi$ a section of $P^{\vee }$ over $S$) by starting from the "universal hyperplane" $H$ and taking its inverse image under the base change $\xi :S\to P^{\vee }$.

The same remark holds for every point of $P^{\vee }$ with values in an arbitrary $S$-prescheme $S^{\prime }$, which (considered as a section of $P_{S^{\prime }}^{\vee }$ over $S^{\prime }$) allows us to define an $H_{\xi }\subset P_{S^{\prime }}$; the latter is nothing else but the inverse image of $H$ by the base change $\xi :S^{\prime }\to P^{\vee }$.

In what follows we assume given a prescheme $X$ of finite type over $P$,⁴ together with an $S$-morphism $f:X\to P$. One of the main objectives of this paragraph is to study, for every hyperplane $H_{\xi }$ of $P$, its inverse image

  Y_ξ = f^{-1}(H_ξ) = X ×_P H_ξ,

and especially to relate the properties of $X$ and $Y_{\xi }$. As usual, one also has to consider the $P^{\vee }(S^{\prime })$ for an arbitrary $S$-scheme $S^{\prime }$; in that case $H_{\xi }$ is a hyperplane in $P_{S^{\prime }}$, and we put again

  Y_ξ = f_{S'}^{-1}(H_ξ) = X_{S'} ×_{P_{S'}} H_ξ = X ×_P H_ξ,

where the subscript $S^{\prime }$ denotes as usual the effect of the base change $S^{\prime }\to S$, and where in the last expression we consider $H_{\xi }$ as a $P$-scheme via the combined morphism $H_{\xi }\to P_{S^{\prime }}\to P$. It is therefore again convenient to consider the case where $\xi$ is "universal", i.e. where $S^{\prime }=P^{\vee }$ and $\xi$ is the diagonal section, so that $H_{\xi }=H$; in this case one observes (subject to better notations to be suggested by Dieudonné) that $Y=Y_{\xi }$. In the general case of a $\xi :S^{\prime }\to P^{\vee }$, one therefore has also $Y_{\xi }=Y{\times }_{P^{\vee }}{S}^{\prime }$. Finally, if $F$ is a sheaf of modules⁵ over $X$, we denote by $G_{\xi }$ its inverse image over $Y_{\xi }$, by $G$ its inverse image over $H$, so that one also has $G_{\xi }=G{\otimes }_{{\mathcal{O}}_{P^{\vee }}}{\mathcal{O}}_{S^{\prime }}$.

Let us summarize in a small diagram the essential constructions and notation.

        F                          G              G_η
        ↓                          ↓                ↓
        X  ⟵  X ×_S P^∨  ⟵        Y    ⟵        Y_η
        ↓        ↓                 ↓                ↓
        P  ⟵  P ×_S P^∨  ⟵        H    ⟵        H_η
        ↓        ↓                 ↘                ↓
        S  ⟵     P^∨           ⟵          ⟵      S'

(The squares and diamonds appearing in this diagram are Cartesian.)

In the next subsection (§V.5.2) we shall study systematically the following case: $S^{\prime }$ is the spectrum of a field $K$, and its image in $P^{\vee }$ is the generic point of the corresponding fibre $P_s^{\vee }$. After making the base change $\mathrm{Spec}k(s)\to S$, we are reduced to the case where $S$ is the spectrum of a field $k$, which is what we shall assume in the next subsection. Also, the majority of properties studied for $X$ and $Y_{\xi }$ are of "geometric" nature and therefore invariant under base change, which allows us also (without loss of generality) to restrict ourselves to the case where $K$ is algebraically closed, or to the case where $K=k(\eta )$, $\eta$ being the generic point of $P^{\vee }$, and $\xi :\mathrm{Spec}(K)\to P^{\vee }$ is of course the canonical morphism. We also note that for geometric questions concerning $X,Y_{\xi }$ we can (after making a base change) restrict ourselves to the case of $k$ algebraically closed.

A terminological reminder. If $f$ is an immersion, we usually call $Y_{\xi }$ a hyperplane section of $X$ (relative to the projective immersion $f$ and the hyperplane $H_{\xi }$). There is no reason not to extend this terminology to the case of an arbitrary $f$.

V.5.2. Study of a generic hyperplane section: local properties

Let us recall that, from now on, $S=\mathrm{Spec}(k)$, with $k$ a field. If $\eta$ is a point of $P^{\vee }$ and if $\xi :\mathrm{Spec}k(\eta )\to P^{\vee }$ is the canonical morphism, we also write $H_{\eta }$, $Y_{\eta }$, $G_{\eta }$ in place of $H_{\xi }$, $Y_{\xi }$, $G_{\xi }$.

In this subsection, $\eta$ always denotes the generic point of $P^{\vee }$.

Proposition (5.2.1).

Assume that $X$ is irreducible. Then $Y_{\eta }$ is irreducible or empty, and in the first case it dominates $X$; the prescheme $Y$ is then irreducible as well.

Indeed, since $H\to P$ is a projective fibration, the same is true of $Y\to X$, which implies that $Y$ is irreducible whenever $X$ is. The generic fibre $Y_{\eta }$ of $Y\to P^{\vee }$ is then irreducible or empty; in the first case its generic point is the generic point of $Y$, which therefore lies over the generic point of $X$.

Proposition (5.2.2).

Let $Z$ be a subset of $P$. Then its inverse image $Z_{\eta }$ in $H_{\eta }$ is empty if and only if every point of $Z$ is closed. In particular, if $Z$ is constructible, then $Z_{\eta }=\varnothing$ if and only if $Z$ is finite.

We may suppose that $Z$ is reduced to a single point $z$, and we only have to prove that the image of $H_{\eta }$ in $P$ consists exactly of the non-closed points of $P$. Denoting by $X$ the closure of $z$ and using (5.2.1), we only have to prove that $Z_{\eta }=\varnothing$ if and only if $X$ is finite ($X$ being a closed subscheme of $P$). Replacing $X$ by $X_{k(\eta )}\hookrightarrow P_{k(\eta )}$, the "only if" part follows from the following fact (for which we need to give a reference and which deserves to be restated here as a lemma): if $Y$ is any hyperplane section of $X$ and if $Y_{\eta }=\varnothing$, then $X$ is finite (indeed, $X\subset P-H$ is affine and projective). The "sufficient" part is immediate: for example, $Y$ is a projective fibration of relative dimension $n-1$ over $X$ ($n$ being the relative dimension of $P$ and $P^{\vee }$ over $S$); since $X$ is finite over $k$, $Y$ is of absolute dimension $n-1=\dim P^{\vee }-1$, hence the morphism $Y\to P^{\vee }$ cannot be dominant and its generic fibre $Y_{\eta }$ is empty.

Corollary (5.2.3).

Let $f:X\to P$ be a morphism of finite type, and let $Z$ be a constructible subset of $X$. In order that its inverse image in $Y_{\eta }$ be empty, it is necessary and sufficient that the image $f(Z)$ be finite. In particular, in order that $Y_{\eta }$ be empty, it is necessary and sufficient that $f(X)$ be finite.

Corollary (5.2.4).

Let Z, Z' be two closed subsets of $X$ with $Z$ irreducible, and let $Z_{\eta }$ and $Z_{\eta }^{\prime }$ be their inverse images in $Y_{\eta }$. In order to have $Z_{\eta }\subset Z_{\eta }^{\prime }$, it is necessary and sufficient that $f(Z)$ be finite, or that $Z\subset Z^{\prime }$. In order that $Z_{\eta }=Z_{\eta }^{\prime }$, it is necessary and sufficient that both $f(Z)$ and $f(Z^{\prime })$ be finite, or that $Z=Z^{\prime }$.

This is an immediate consequence of (5.2.3), since we see that $f(Z-Z\cap Z^{\prime })$ can be finite only if $Z\subset Z^{\prime }$ or if $f(Z)$ is finite: if $Z\not\subset Z^{\prime }$, then $Z-Z\cap Z^{\prime }$ is dense in $Z$, so $f(Z-Z\cap Z^{\prime })$ is dense in $f(Z)$; if the former is finite and hence (being constructible) closed, then so is the latter.⁶

Corollary (5.2.5).

To every irreducible component $X_i$ of $X$ such that $\dim f(X_i)>0$ we assign its inverse image $Y_{i,\eta }$ in $Y_{\eta }$. Then $Y_{i,\eta }$ is an irreducible component of $Y_{\eta }$, and we obtain in this manner a one-to-one correspondence between the set of irreducible components $X_i$ of $X$ such that $\dim f(X_i)>0$ and the set of irreducible components of $Y_{\eta }$.

Indeed, it follows from (5.2.3) that $Y_{\eta }$ is the union of the $Y_{i,\eta }$ defined above, and that the latter are closed and non-empty subsets of $Y$; they are also irreducible by (5.2.1). Finally, they are mutually without an inclusion relation by (5.2.4), whence the conclusion.

Let us notice that if $\dim X_i=d_i$, then $\dim Y_{i,\eta }=d_i-1$. More generally:

Proposition (5.2.6).

Suppose that for every irreducible component $X_i$ of $X$ we have $\dim f(X_i)>0$, i.e. $Y_{i,\eta }\ne \varnothing$; or, where $f$ is an immersion, that $\dim f(X)>0$. Then $\dim Y_{\eta }=\dim X-1$.

We are reduced to the case where $X$ is irreducible, by (5.2.5). By the very construction, $Y_{\eta }$ is defined starting from $X_{k(\eta )}$ as the divisor of a section of an invertible module over $X_{k(\eta )}$ (the inverse image of ${\mathcal{O}}_P(1)$). On the other hand, $X_{k(\eta )}$ is irreducible (because $X$ is such and $k(\eta )$ is a purely transcendental extension of $k$, which fact one should have indicated at the beginning of the subsection), and $Y_{\eta }\ne X_{k(\eta )}$, since the image of $Y_{\eta }$ in $X$ (contrary to that of $X_{k(\eta )}$, which is faithfully flat over $X$) is not equal to $X$: indeed, it does not contain the closed points of $X$, by (5.2.3). It follows that dim Y_η = dim X_{k(η)} − 1 = dim X − 1.

Proposition (5.2.7).

Let $F$ be a quasi-coherent module over $X$, and hence $G_{\eta }$ over $Y_{\eta }$. Let $(Z_i)$ be the prime cycles associated to $F$ such that $\dim f(Z_i)>0$, and let $Z_{i,\eta }$ be the inverse image of $Z_i$ in $Y_{\eta }$. Then the $Z_{i,\eta }$ are exactly all the prime cycles associated to $G_{\eta }$. Moreover, their inclusion relations are the same as those among the $Z_i$.

The last assertion is contained in (5.2.4). On the other hand, since $Y\to X$ is a projective fibration — hence flat with fibres (S_1) and irreducible — it follows from §IV.3 that the prime cycles associated to the inverse image $G$ of $F$ over $Y$ are the inverse images of the prime cycles associated to $F$. Restricting to the generic fibre $Y_{\eta }$ of $Y$ over $P^{\vee }$, we obtain that the prime cycles associated to $G_{\eta }$ are the non-empty inverse images of the $Z_i$, which proves (5.2.7) by means of (5.2.3).

To tell the truth, the passage through $Y$ is unnecessary: we can use directly the fact that $Y_{\eta }\to X$ is flat with fibres (S_1) and irreducible (in fact even geometrically regular, and with geometrically irreducible fibres, the latter being localizations of projective schemes); this is the remark to make for the proof of (5.2.1).

Proposition (5.2.8).

Let $F$ be coherent over $X$, let $y\in Y_{\eta }$, and let $x$ be its image in $X$. Let $P(M)$ be one of the following properties of a module of finite type $M$ over a local noetherian ring $A$:

(i) $coprofM\le n$;

(ii) $M$ satisfies $(S_k)$;

(iii) $M$ is Cohen-Macaulay;

(iv) $M$ is reduced;

(v) $M$ is integral.

Then in order that $G_{\eta ,y}$ should satisfy $P$, it is necessary and sufficient that $F_x$ should satisfy it.

This follows immediately from the results of §IV.6, taking into account that $Y_{\eta }\to X$ is a regular morphism, so that ${\mathcal{O}}_{X,x}\to {\mathcal{O}}_{Y_{\eta },y}$ is regular.⁷ Taking (5.2.3) into account, we obtain:

Corollary (5.2.9).

With the notation of (5.2.8), let $Z$ be the set of $x\in X$ such that $P(F_x)$ is not satisfied. Then in order that $G_{\eta }$ should satisfy the condition $P$ at every point, it is necessary and sufficient that $f(Z)$ be a finite subset of $P$, or — equivalently — that $\dim f(Z)=0$.

Indeed, (5.2.8) says that $h^{-1}(Z)$ is the $P$-singular subset of $G_{\eta }$, and it is empty if and only if $f(Z)$ is finite, by (5.2.3). (Here $h$ denotes the morphism $Y_{\eta }\to X$. I have just realized that the letter $P$ in (5.2.8) has been used twice; this should be resolved at the editing stage.)

Corollary (5.2.10).

The analogous condition for $Y_{\eta }$ to be reduced, respectively locally integral, follows from (5.2.8) (iv), (v).

Corollary (5.2.11).

Let $y\in Y_{\eta }$. In order that $Y_{\eta }$ should be regular, respectively $(R_k)$ at $y$ (respectively normal at $y$), it is necessary and sufficient that $X$ should satisfy the same property at $x$. Let $Z$ be the set of points of $X$ where $X$ is not regular, respectively not $(R_k)$, respectively not normal; for $Y_{\eta }$ to be regular, respectively $(R_k)$, respectively normal, it is necessary and sufficient that $f(Z)$ be finite, i.e. that $\dim f(Z)=0$.

The proof is the same as for (5.2.8) and (5.2.9). One must give the various references assuring that $Z$ is closed (since one needs to know it is constructible in order to apply (5.2.3)).

Let us note that in (5.2.10) and (5.2.11) we do not speak at all of the corresponding geometric properties; the results described are of "absolute" nature. We now examine the properties of geometric nature. (One could, if one wished, take the opportunity to start a new subsection here.)

Geometric properties

Theorem (5.2.12).

Suppose that $f:X\to P$ is unramified. Let $y\in Y_{\eta }$, and let $x$ be its image in $X$. In order that $X$ should be smooth over $k$ at $x$, it is necessary and sufficient that $Y_{\eta }$ should be smooth over $k(\eta )$ at $y$.

We may assume that $k$ is algebraically closed. If $Y_{\eta }$ is smooth over $k(\eta )$ at $y$, it is regular, and so, since $Y_{\eta }$ is flat over $X$, $X$ is regular at $x$; therefore $X$ is smooth over $k$ at $x$, since $k$ is algebraically closed and thus perfect.

For the converse, we can (after replacing $X$ by an open neighbourhood of $x$) assume that $X$ is smooth, and, by the Jacobian criterion of smoothness, that it is defined in an open subset $U$ of $P$ by $p$ equations,

$$X=V(f_1,\cdots ,f_p),$$

where the differentials $d{f}_1,\cdots ,d{f}_p$ are everywhere linearly independent. Introducing the affine coordinates $S_1,\cdots ,S_n$ in $P^{\vee }$ and the affine coordinates $T_1,T_2,\cdots ,T_n$ in a neighbourhood of $x$ (by choosing a hyperplane $H_{\infty }$ at infinity not containing $x$), the immersion $Y_{\eta }\hookrightarrow U_{k(\eta )}$ is then given by

  Y_η = V(f_1, …, f_p, ∑ S_i T_i − 1),

and it suffices to verify that the differentials (relative to $k(\eta )$) of $f_1,\cdots ,f_p,\sum S_i{T}_i-1$ are linearly independent. These differentials are nothing else but the sections

  df_1, …, df_p, ∑ S_i dT_i

of ${\Omega }_{U/k}^1{\otimes }_{k}k(\eta )$. Since the $d{f}_i$ are linearly independent at every point of $U$, and since the $d{T}_i$ form a basis of ${\Omega }_{U/S}^1$ at every point of $U$ (and a fortiori a system of generators), we conclude immediately the linear independence of the displayed quantities at every point of $U_{k(\eta )}$ — at least when $p<n$, i.e. when

  E = Ω^1_{U/k} / ∑_{1 ≤ i ≤ p} 𝒪_U df_i ≠ 0.

This is a small lemma about a family of generators $a_i$, $1\le i\le n$, of a non-zero locally free module $E$: the section $\sum S_i{a}_i$ of $E{\otimes }_{k}k(\eta )$ does not vanish at any point. On the other hand, the case $p=n$ is trivial, because then $Y_{\eta }=\varnothing$.

Corollary (5.2.13).

Let $Z$ be the set of points of $X$ where $X$ is not smooth over $k$. In order that $Y_{\eta }$ should be smooth over $k(\eta )$, it is necessary and sufficient that $Z$ be finite. In particular, if $X$ is smooth, so is $Y_{\eta }$.

This follows from (5.2.12) and (5.2.3). More generally we obtain:

Theorem (5.2.14).

Let $y$ be a point of $Y_{\eta }$, and $x$ its image in $X$. Let $P(A,K)$ be one of the following properties of a local noetherian $K$-algebra $A$ over a field $K$:

(i) $A$ is geometrically regular over $K$;

(ii) $A$ is geometrically $(R_k)$ over $K$;

(iii) $A$ is separable over $K$;

(iv) $A$ is geometrically normal over $K$.

Then $P({\mathcal{O}}_{X,x},k)⟺P({\mathcal{O}}_{Y_{\eta },y},k(y))$.

Indeed, taking §IV.6 into account, (iii) and (iv) follow from (ii) and (5.2.8) (ii). On the other hand, (i) has been proven in (5.2.12), and it remains to deduce (ii) from (i). To do this, let $Z$ be the set of points where $X$ is not smooth over $k$; its inverse image $Z_{\eta }$ in $Y_{\eta }$ is therefore (by (5.2.12)) the set of points of $Y_{\eta }$ at which $Y_{\eta }$ is not smooth over $k(\eta )$. But the codimension of $Z$ in $X$ equals that of $Z_{\eta }$ in $Y_{\eta }$ at $y$ because of flatness (see §IV.6); therefore one is $>k$ if and only if the other is, which completes the proof.

Corollary (5.2.15).

Let $Z$ be the set of points of $X$ at which $X$ is not smooth over $k$ (respectively is not geometrically $(R_k)$ over $k$, respectively is not separable over $k$, respectively is not geometrically normal over $k$). In order that $Y_{\eta }$ should be smooth (respectively geometrically $(R_k)$, respectively separable, respectively geometrically normal) over $k(\eta )$, it is necessary and sufficient that $Z$ be finite.

From the point of view of presentation, statements (5.2.14) and (5.2.15) contain (5.2.12) and (5.2.13) (which we could thus dispense with stating separately); on the other hand, the corollary is practically more important than the theorem, which one could give in a proposition or a preliminary lemma so that the corollary is the more glorified.

We can give a variant in the case of property (iii):

Corollary (5.2.16).

Suppose that $f$ is an immersion and that $F$ is coherent. Under the conditions of (5.2.7), in order that $Z_i$ should not be embedded, it is necessary and sufficient that $Z_{i,\eta }$ should not be embedded. If that is so, then the radical multiplicity of $F$ at $Z_i$ relative to $k$ equals that of $G_{\eta }$ at $Z_{i,\eta }$ relative to $k(\eta )$.

The first assertion is contained in the last assertion of (5.2.7). For the second, since $Y_{\eta }\to X$ is flat, the functor $F\mapsto G_{\eta }$ is exact, and we are reduced by restriction to a neighbourhood of the generic point of $Z_i$ and by dévissage to the case where $F={\mathcal{O}}_{Z_i}$ and we may assume $Z_i=X$. Also, we could start by assuming that $X$ is separated over $k$, and reduce to the case of $k$ algebraically closed.⁸ Then the assertion is contained in (5.2.15) (iii). We conclude, as usual:

Corollary (5.2.17).

Let $Z$ be the set of points of $X$ where $F$ is not separable over $k$. Then $G_{\eta }$ is separable over $k(\eta )$ if and only if $Z$ is finite. In particular, if $F$ is separable over $k$, then $G_{\eta }$ is separable over $k(\eta )$.

Remark (5.2.18).

The key result (5.2.12) (and its consequences (5.2.13) and (5.2.17)) become false if we abandon the assumption that $f$ is an immersion, as we see for example by taking for $X$ a purely inseparable covering of $P$. However, if $k$ is of characteristic zero, (5.2.12) and (5.2.17) are valid without assuming that $f$ is an immersion.

Indeed, it suffices to verify this for (5.2.12), and this follows from (5.2.11) and the fact that, for an algebraic prescheme in characteristic zero, smooth = regular.⁹

V.5.3. Generic hyperplane section: geometric irreducibility and connectedness

Theorem (5.3.1) (Bertini-Zariski).

Assume that $\dim f(X)\ge 2$ and that $X$ is geometrically irreducible. Then the generic hyperplane section $Y_{\eta }$ has the same property.

Let $K/k$ be the function field of $X$, and let $n=\dim P$; introducing the affine coordinates $T_1,\cdots ,T_n$ in $P$ (by choosing a hyperplane at infinity $H_{\infty }$ such that $f(X)$ is not contained in it) and $S_1,\cdots ,S_n$ the affine coordinates in $P^{\vee }$, we see that the function field $L$ of $Y_{\eta }$ can be identified with the field of fractions of the integral domain $K[S_1,\cdots ,S_n]/(\sum t_i{S}_i-1)$, where the $t_i\in K$ are the images of $T_i$ under $f:X\to P$. Since $\dim f(X)>0$, the $t_i$ are not all algebraic over $k$ — a fortiori, they are not all zero; suppose, for example, that $t_n\ne 0$. We realize then immediately that

$$L=K(S_1,\cdots ,S_{n-1})$$

(a purely transcendental extension), $S_n\in L$ being given by the equation $\sum t_i{S}_i-1=0$ as a function of the $S_i$ ($1\le i\le n-1$) and the $t_i$ ($1\le i\le n$). On the other hand, $k^{\prime }=k(\eta )$ can be identified with $k(S_1,\cdots ,S_n)$, and the canonical inclusion $k^{\prime }\to L$ is obtained by sending each $S_i$ to its image in $L$;¹⁰ that is, $k^{\prime }$ as a subextension of $L$ is generated by the $S_i$ ($1\le i\le n$), or, what is the same, by the $S_i$ ($1\le i\le n-1$) together with

  S_n = a_0 + a_1 S_1 + ⋯ + a_{n-1} S_{n-1},

where $a_0=t_n^{-1}$ and $a_i=-t_i{t}_n^{-1}$ for $1\le i\le n-1$.

Note that the field generated by the $a_i$ and that generated by the $t_i$ are obviously the same; their common transcendence degree is nothing else but the dimension of $f(X)$.

(N.B. It would be appropriate to include this birational description at least as a corollary to (5.2.1).)

The proof of (5.3.1) is thus reduced to that of:

Lemma (5.3.1.1) (Zariski).

Let $k$ be a field, $K$ an extension of finite type over $k$, $m$ an integer $\ge 0$, and $a_i$ ($0\le i\le m$) elements of $K$ such that the transcendence degree of $k(a_0,\cdots ,a_m)$ over $k$ is $\ge 2$. Let $L=K(S_1,\cdots ,S_m)$ and

let $k^{\prime }$ be the subfield

  k' = k(S_1, …, S_m, a_0 + ∑_{1 ≤ i ≤ m} a_i S_i)

of $L$, the $S_i$ being indeterminates. If $K$ is a primary extension of $k$, then $L$ is a primary extension of $k^{\prime }$.¹¹

This lemma, or lemmas that resemble it like a brother, wander almost everywhere in the literature. That is why I leave it up to you: the choice of the place from which you will copy a proof. I do not feel inspired to find a proof with my own means.

Corollary (5.3.2).

Assume that $f$ is unramified, or that $k$ is of characteristic zero, and that $\dim f(X)\ge 2$. Then if $X$ is geometrically integral, the same is true of $Y_{\eta }$.

Indeed, geometrically integral = geometrically irreducible + separable.

Corollary (5.3.3).

Assume that $k$ is algebraically closed and that for every irreducible component $X_i$ of $X$ we have $\dim f(X_i)\ge 2$. Suppose also that $X$ is $\sigma$-connected, where $\sigma$ is the set of closed subsets $Z$ of $X$ such that $\dim f(Z)=0$ (i.e. for every such $Z$, $X-Z$ is connected). Under these conditions, $Y_{\eta }$ is geometrically connected over $k(\eta )$.

Indeed, by a lemma that ought to appear in §IV.6 together with Hartshorne's theorem,¹² the hypothesis signifies that we can join any two irreducible components $X^{\prime }$ and X'' of $X$ by a chain of irreducible components $X_0=X^{\prime },X_1,\cdots ,X_n=X^{\prime \prime }$ such that two consecutive ones have an intersection not in $\sigma$; consequently the inverse images $X_{i,\eta }$ are joined by a chain of components which are geometrically connected over $k(\eta )$ by (5.3.1) and have pairwise non-empty intersection by (5.2.3).

It follows (since $Y_{\eta }=X_{\eta }$ is the union of the $X_{i,\eta }$ as $X_i$ runs through the irreducible components of $X$) that $Y_{\eta }$ is geometrically connected over $k(\eta )$.

Translator's note to (5.3.1.1). This lemma should be compared with Zariski's collected papers (MIT Press), vol. 1, page 174, and vol. 2, page 304; also Zariski-Samuel, vol. 1, page 196, and vol. 2, page 230 of the GTM Springer edition. See also Jouanolou's Théorème de Bertini et applications, Theorem 3.6 and Section 6.

V.5.4. Variable hyperplane section: "sufficiently general" sections

We return to the general situation of §V.5.1: $S$ an arbitrary prescheme. We also suppose that $X$ is of finite presentation over $S$.

In general, let us note that if $P(Z,k)$ is a "constructible" property of an algebraic prescheme $Z$ over a field $k$, then the set of $\xi \in P^{\vee }$ such that $P(Y_{\xi },k(\xi ))$ holds is constructible. Indeed, $Y_{\xi }$ is the fibre over $\xi$ of $Y\to P^{\vee }$, which is a morphism

of finite presentation, and we apply §IV.9. We have an analogous remark for a property $P(Z,M,k)$, where $Z$ and $k$ are as above and $M$ is a coherent module over $Z$: if $G$ is in addition of finite presentation over $X$, then the set of $\xi \in P^{\vee }$ such that $P(Y_{\eta },G_{\eta },k(\eta ))$ holds¹³ is constructible. On the other hand, in the previous two subsections we have developed, in various cases, criteria for the set $E$ above to contain the generic point of $P^{\vee }$, $S$ being the spectrum of a field $k$; being constructible, it follows that $E$ contains a non-empty open set: this is the passage from a conclusion concerning the generic hyperplane section to the analogous conclusion for "sufficiently general" hyperplane sections.

Let us note in addition that, in the case $S=\mathrm{Spec}(k)$, we also have a converse: in order that the generic hyperplane section should have the property $P$, it is necessary and sufficient that the $Y_{\xi }$ for $\xi$ in a suitable neighbourhood of $\eta$ should satisfy it; and it suffices to require this for $\xi$ closed in $P^{\vee }$, which for $k$ algebraically closed leads to (or reduces to) considering $k$-rational points, i.e. hyperplane sections of $X$ itself (without a prior extension of the base field).¹⁴

This follows, indeed, from the constructibility of $E$ and from the fact that $P^{\vee }$ is a Jacobson scheme.

Let us give as examples some special cases where the previous considerations apply.

Proposition (5.4.2).¹⁵

Let $G$ be a module of finite presentation over $X$. Let $P$ be one of the following properties for a module $M$ over an algebraic scheme $Z$ over a field $K$:

(i) $coprof(M)\le n$;

(ii) $M$ satisfies $(S_k)$;

(iii) $M$ is Cohen-Macaulay;

(iv) $M$ has no embedded prime cycle components;

(v) $M$ is separable over $K$.

With these notations, if $E$ denotes the set of $\xi \in P^{\vee }$ such that $G_{\xi }$ satisfies the property $P$, then:

(a) $E$ is a constructible subset of $P^{\vee }$.

(b) Suppose that $S=\mathrm{Spec}(k)$, $k$ a field, and that $F$ satisfies the property $P$. Suppose also, in case (v), that $k$ is of characteristic zero, or that $f:X\to P$ is unramified. Then $E$ contains an open dense subset of $P^{\vee }$.

Proof. (a) follows from the fact that $P$ is a constructible property, as we have seen in §IV.9. As to (b), it follows from the corresponding results of the previous two subsections.

Remark. More generally one could suppose that, with $Z$ the set of points of $X$ where $F$ does not satisfy $P$, the image $f(Z)$ is finite, i.e. $\dim f(Z)\le 0$.

Proposition (5.4.3).

Let $P$ be one of the following properties (for an algebraic prescheme $Z$ over a field $K$):

(i) $Z$ is smooth over $K$;

(ii) $Z$ satisfies the geometric property $(R_k)$ over $K$;

(iii) $Z$ is separable over $K$;

(iv) $Z$ is geometrically normal over $K$;

(v) $Z$ is geometrically integral over $K$;

(vi) $Z$ is geometrically irreducible over $K$.

Let $E$ be the set of $\xi \in P^{\vee }$ such that $Y_{\xi }$ satisfies $P$. Then:

(a) $E$ is a constructible subset of $P^{\vee }$.

(b) Suppose $S=\mathrm{Spec}(k)$, $k$ a field, and suppose in cases (i)-(v) that $k$ is of characteristic zero and that $f:X\to P$ is unramified. Finally, suppose in cases (v) and (vi) that $\dim f(X)\ge 2$. If $X$ satisfies $P$, then $E$ contains a dense open subset of $P^{\vee }$.

Proof. The proof is identical to that of (5.4.2). Note that in cases (i)-(v) it suffices to assume only (in (b)) that $f(Z)$ is finite, where $Z$ is the set of points of $X$ where $P$ fails.

Corollary (5.4.4).

Consider the property $(C_m)$: "$\overline{Z}$ cannot be disconnected by a closed subset of dimension $\le m$", where $\overline{Z}=Z{\otimes }_K\overline{K}$, $\overline{K}$ the algebraic closure of $K$. Let $E$ be the set of $\xi \in P^{\vee }$ such that $Y_{\xi }$ over $k(\xi )$ satisfies $(C_m)$. Then:

(a) $E$ is constructible.

(b) Suppose $S=\mathrm{Spec}(k)$, $k$ a field, and that for every irreducible component $X_i$ of $X$ we have $\dim f(X_i)\ge 2$. Then if $X$ over $k$ satisfies $(C_{m+1})$, then $E$ contains a dense open subset $U$ of $P^{\vee }$.

The constructibility is done via "AQT".¹⁶ This is a fact that one has forgotten in §IV.9 — perhaps it would still be possible to repair (or fix) it. Part (b) follows in principle from (5.3.3), except that (5.3.3) has been stated in an excessively special manner and consequently should be generalized (the proof being otherwise essentially unchanged).

Also, there is an error in the statement of (5.4.4): it is not valid as such if $f$ is quasi-finite. In the general case, instead of considering the dimension of the closed subsets of $X$, respectively of $Y_{\xi }$, it is sufficient to consider the dimension of their images in $P$, respectively in $H_{\xi }$. Let the redactor sort himself out.¹⁷

V.5.5. Theorems of Seidenberg type

(5.5.1). In the present subsection we give conditions under which the set $E$ defined in §V.5.4 is open. We deal here with properties $P$ of local nature over $X$, respectively $Y_{\xi }$, such that we can define the set $U$ of $y\in Y$ for which ($\xi$ being the image of $y$ in $P^{\vee }$) $Y_{\xi }$ satisfies $P$ at the point $y$ (respectively $G_{\xi }$ satisfies the condition $P$ at $y$). We give first the criteria for $U$ to be open, by using §IV.12.¹⁸ As always we have $E=P^{\vee }-h(Y-U)$. It follows that if $U$ is open and $X$ is proper over $S$ (since $h$ is then proper and a fortiori closed), then $E$ is also open.¹⁹

(5.5.2). As we have seen in §V.5.1, $Y$ is defined in $X{\times }_S{P}^{\vee }=X_{P^{\vee }}$ as the "divisor" of a section $\varphi$ of ${\mathcal{O}}_X(1){\otimes }_S{\mathcal{O}}_{P^{\vee }}(1)$, which induces for every $\xi \in P^{\vee }$ a section ${\varphi }_{\xi }$ of ${\mathcal{O}}_X(1){\otimes }_{k(s)}{\mathcal{O}}_{P^{\vee }}(1)(\xi )$ (a sheaf, by the way, isomorphic non-canonically to ${\mathcal{O}}_X(1){\otimes }_{k(s)}k(\xi )={\mathcal{O}}_{X_{k(s)}}(1)$), such that $Y_{\xi }$ is nothing else but the "divisor" of this section. (N.B. The divisor of a section $\varphi$ of an invertible module $L$ is defined as the closed subscheme defined by the image ideal of ${\varphi }^{-1}:L^{-1}\to \mathcal{O}$.) If $F$ is a sheaf of modules over $X$, then its inverse image over $Y$, i.e. the inverse image of $F{\otimes }_{{\mathcal{O}}_S}{\mathcal{O}}_{P^{\vee }}=F_{P^{\vee }}$ over the subscheme $Y$ of $X_{P^{\vee }}$, is nothing else but the cokernel of the homomorphism

  (φ ⊗ id_{F_{P^∨}})^{-1} : F_{P^∨}(-1, -1) ⟶ F_{P^∨},

where the notation (-1, -1) explains itself (as M. Artin says²⁰). Also, $G_{\xi }$ is the cokernel of the analogous homomorphism

$$F_{k(\xi )}(-1,-1)⟶F_{k(\xi )},$$

where $\xi$ is a point of $P^{\vee }$ (with a corresponding interpretation if $\xi$, instead of being a point of $P^{\vee }$, denotes a point of $P^{\vee }$ with values in an $S^{\prime }$ over $S$).

In general, if $L$ is an invertible module on some scheme and $\varphi$ is a section defining the subprescheme $V(\varphi )$, then for every module $F$ the inverse image of $F$ in $V(\varphi )$ can be identified, by the usual abuse of language, with the cokernel of $i{d}_F\otimes \varphi :F\otimes L^{-1}\to F$.

We say that $\varphi$ is $F$-regular if the preceding homomorphism is injective. If we choose locally an isomorphism of $L$ with ${\mathcal{O}}_X$ — which is possible — so that $\varphi$ is identified with a section of ${\mathcal{O}}_X$, this terminology is compatible with the one already introduced elsewhere.

Proposition (5.5.3).

With the previous notation, let $U$ be the set of $x\in X_{P^{\vee }}$ with image $\xi$ in $P^{\vee }$ such that ${\varphi }_{\xi }$ is $F_{k(\xi )}$-regular at $x$. Then:

(a) If $F$ is of finite presentation and flat relative to $S$, then $U$ is open and $G|U$ is flat relative to $P^{\vee }$.

(b) For every $s\in S$, if $\eta$ denotes the generic point of $P_s^{\vee }$, then $U$ contains $X_{k(\eta )}$.

Proof.

(a) Since $F_{P^{\vee }}$ is of finite presentation and flat relative to $P^{\vee }$, the conclusion follows from §IV.11.3 (and also from §0_III, in the case of locally noetherian $S$).

(b) We may suppose $S=\mathrm{Spec}(k)$. The associated cycles of $F_{k(\eta )}$ are (by §IV.3) the inverse images of the associated cycles $Z_i$ of $F$. If $f(Z_i)$ is finite, then by (5.2.3) $Z_{i,k(\eta )}\cap Y=\varnothing$; in the contrary case, by (5.2.6) (or — better — by reasons of dimension; (5.2.3), already invoked in (5.2.6), is undoubtedly a better reason), we have $Z_{i,k(\eta )}\cap Y=Z_{i,k(\eta )}$. This proves that $\varphi$ does not vanish on any of the $Z_{i,k(\eta )}$ and therefore proves (b).

Corollary (5.5.4).

Let $V$ be the set of $\xi \in P^{\vee }$ such that ${\varphi }_{\xi }$ is $F_{k(\xi )}$-regular. If $F$ is of finite presentation, then $V$ is constructible and it contains the generic points of the fibres of $P^{\vee }$ over $S$. On the other hand, if also $X$ is proper over $S$ and $F$ is flat over $S$, then the set $V$ is open.

Remark (5.5.5).

Let $\xi \in P^{\vee }$ over $s\in S$, and suppose that $F_s$ is without associated embedded cycles. Then $\xi \in V$ (notation of (5.5.4)) — which means also that every irreducible component of $supp{F}_{k(\xi )}$ does not lie over $H_{\xi }$ (and somewhat less evidently, in this criterion we may replace $k(\xi )$ by an arbitrary extension of $k(\xi )$).

Let us note that the hypothesis (S_1) about $F_s$ just made is satisfied notably if $F_s$ is Cohen-Macaulay (a fortiori if $F$ is CM over $S$); also in this case $G_s$ is CM (since locally it is deduced from $F_{k(s)}$, which is such, by dividing by $\varphi \cdot F_{k(s)}$ where $\varphi$ is $F_{k(s)}$-regular). The same remarks should (and will have to) be made locally above to characterize the points of $U$ (in place of those of $V$).

Using now §§IV.12.1.1 and IV.12.1.4, we obtain:

Theorem (5.5.6).

Assume that $F$ is of finite presentation and flat relative to $S$. Let $P$ be one of the properties (i)-(viii) of §IV.12.1.1, or (if we assume $F={\mathcal{O}}_X$) one of the properties (i)-(iv) of §IV.12.1.4.²¹ Let U_P be the set of $x\in X_{P^{\vee }}$ such that, if $\xi$ denotes the image of $x$ in $P^{\vee }$, the property $P$ is satisfied by $G_{\xi }$ (resp. $Y_{\xi }$) at the point $x$, and such that ${\varphi }_{\xi }$ is $F_{k(\xi )}$-regular at $x$. Then U_P is open and $G|U_P$ is flat relative to $S$.

Indeed, by the very definition we have $U_P\subset U$ (notation of (5.5.3) (a)), and we apply §IV.12 to $U\to P^{\vee }$ and $F_{P^{\vee }}|U$.

Corollary (5.5.7).

Suppose that $F$ is of finite presentation and flat relative to $S$, and that supp F is proper over $S$ (e.g. $X$ proper over $S$). Let V_P be the set of $\xi \in P^{\vee }$ such that $G_{\xi }$

(resp. $Y_{\xi }$) satisfies the property $P$ and ${\varphi }_{\xi }$ is $F_{k(\xi )}$-regular. Under these conditions, V_P is open (and is constructible in every case, even without any flatness or properness assumption).

It seems to me that from the point of view of presentation we cannot leave (5.5.6) as is with a simple reference to the conditions enumerated in another volume; it requires an explicit list (i), (ii), … of properties which we have in view. Remark also (in (5.5.1) perhaps) that the case $P=$ geometrically normal (with $S=\mathrm{Spec}(k)$, to be sure²²) is due to Seidenberg.

V.5.6. Connectedness of an arbitrary hyperplane section

We now combine the already-known criterion for geometric connectedness of the generic hyperplane section (5.3.3) with Zariski's connectedness theorem in order to obtain a connectedness result for an arbitrary hyperplane section.

Proposition (5.6.1).

Suppose $S=\mathrm{Spec}(k)$, $k$ an algebraically closed field, $X$ proper over $k$. Suppose that for every irreducible component $X_i$ of $X$, $f(X_i)$ should be of dimension $\ge 2$, and finally that $X$ cannot be disconnected by a closed subset $Z$ of $X$ such that $\dim f(Z)\le 0$. Under these conditions, for every $\xi \in P^{\vee }$, $Y_{\xi }$ is geometrically connected.

Proof. Since none of the $f(X_i)$ is finite, we see that every irreducible component $Y_i$ of $Y$ dominates $P^{\vee }$. On the other hand, $Y\to P^{\vee }$ is proper ($Y$ being proper over $k$, since $Y$ is proper over $X$ and $X$ is proper over $k$). On the other hand, by (5.3.3), the generic fibre $Y_{\eta }$ of $Y\to P^{\vee }$ is geometrically connected.

Finally, $P^{\vee }$ is regular and a fortiori geometrically unibranch. It now suffices to apply §IV.15.6.3 (which is a variant of Zariski's connectedness theorem) to conclude that all the fibres of $Y\to P^{\vee }$ are geometrically connected.

Grothendieck note. It is not difficult, by a proof of analogous type, to generalize (5.6.1) in the same sense as (5.4.4). If you do not want to trouble yourself with this exercise, at least mention it as a remark. Note also that we do not discriminate in (5.6.1) with regard to hyperplane sections that have an excessive (extra) dimension. From the planning point of view, it might be clearer to group together all the connectedness questions (including (5.3.3) and (5.4.4)) in the same subsection.

V.5.7. Application to the construction of hyperplane sections and multisections of specified type

(5.7.1). Let us note that if $S=\mathrm{Spec}(k)$ where $k$ is an infinite field, then every non-empty open subset of $P^{\vee }$ contains a $k$-rational point; therefore, with the notation of §V.5.4, if $E$ (defined in terms of a constructible property $P$) contains the generic point $\eta$, it contains a $k$-rational point, and hence there exists a hyperplane section of $X$ (defined over $k$) having the property $P$. On the other hand, $S$ being again arbitrary, it is immediate that for every $s\in S$ and for every point ${\xi }_0$ of the fibre $P_s^{\vee }$ rational over $k(s)$, there exists a section $\xi$ of $P^{\vee }$ on an open neighbourhood $U$ of $s$ which passes through ${\xi }_0$. If $E$ is again defined as in §V.5.4 in terms of a constructible property $P$, and if we have (for example by §V.5.5) that $E$ is open, then if ${\xi }_0\in E$, the section $\xi$ is a section of $E$ over $U$, at least if we sufficiently shrink $U$. Therefore we may construct a hyperplane section $Y_{\xi }$ of $X$ over an open neighbourhood $U$ of $s$ such that for every $t\in U$ its fibre $Y_{\xi (t)}$ at $t$ satisfies the property $P$. If we do not have a priori ${\xi }_0$ but if $k(s)$ is infinite, we may combine the two preceding remarks to obtain a hyperplane section over an open neighbourhood of $s$ having the preceding property.

Finally, using §V.5.5, we have a criterion allowing us to assert ($X$ resp. $F$ being assumed flat over $S$, which allows us to apply the cited result) that $Y_{\xi }$ resp. $G_{\xi }$ is also flat over $S$. We may therefore, replacing $X$ by $Y_{\xi }$, iterate the previous construction; this allows, under certain conditions, to construct "by successive approximations"²³ a multisection $S^{\prime }$ of $X$ over an open neighbourhood $U$ of the given point $s$, such that $S^{\prime }\to U$ is finite, flat, and with fibres satisfying the property $P$.

If $k(s)$ is finite, we may be forced to do an étale and surjective base change $S^{\prime \prime }\to U$ ($U$ an open neighbourhood of $s$) before being able to apply the preceding constructions. Indeed, under the conditions from the start of §V.5.6, if $k$ is finite, there does not necessarily exist a rational point over $k$ in the open non-empty set $U$; but there certainly exists a closed point of $U$, hence a point with values in a finite extension $k^{\prime }$ (necessarily separable) of $k$. When $k=k(s)$, we may therefore, after making a suitable finite étale extension S'' over a neighbourhood $U$ of $s$, corresponding to the residual extension $k^{\prime }$ (i.e. such that $S_{s^{\prime }}^{\prime \prime }\cong \mathrm{Spec}(k^{\prime })$), restrict ourselves to the favourable situation of the unique point $s^{\prime }\in S^{\prime \prime }$ over $s$ after the base change $S^{\prime \prime }\to S$.

Grothendieck note. I must, however, note a regret regarding (5.4.2) and (5.4.3), which should have been announced in a slightly more general form (at least as a remark). If we are given an integer $m$ and we denote by $E$ the set of $\xi \in P^{\vee }$ such that $G_{\xi }$, resp. $Y_{\xi }$, satisfies $P$ except over a closed set of dimension $\le m$ (i.e. the $P$-singular set $Z$ is of dimension $\le m$), then:

(a) $E$ is a constructible subset of $P^{\vee }$; and

(b) in the case $S=\mathrm{Spec}(k)$, if $F$, respectively $X$, satisfies $P$ except over a set of dimension $\le m+1$, then $E$ contains a non-empty open set.

Proposition (5.7.2).

Assume that $X$ is proper over $S$ and that $F$ is of finite presentation and flat over $S$. Let $P$ be one of the properties (i)-(v) of (5.4.2), and let $m$ be an integer. Let $s\in S$, and suppose that the set $Z_s$ of points of $X_s$ where $F_s$ does not satisfy $P$ is of dimension $\le m+1$. Then if also $k(s)$ is infinite, there exists a neighbourhood $U$ of $s$ in $S$ and a section $\xi$ of $P^{\vee }$ over $U$ having the following properties: for every $x\in U$, the set of points of $Y_{\xi ,x}$ where $G_{\xi (x)}$ does not satisfy $P$ is of dimension $\le m$, and ${\varphi }_{\xi (x)}$ is $F_{\xi ,x}$-regular. Under these conditions, the module $G_{\xi }$ over $Y_{\xi }$ is flat relative to $U$. Finally, if $k(s)$ is not assumed infinite, we can again make the previous construction after an étale extension of the type anticipated in (5.7.1).

Proposition (5.7.3).

The same as (5.7.2), but with no module $F$, assuming instead that $X$ is flat relative to $S$. We refer to properties (i)-(v) of (5.4.3) in place of those of (5.4.2) (but being careful to keep the reservation that in case (v), for every $s\in S$ and every irreducible component $Z$ of $X_s$, we have $\dim f(Z)\ge 2$).²⁴

(Text crossed out in the source.)

Proposition (5.7.4).

Let $g:X\to S$ be a flat proper morphism, let $s\in S$, put $n=\dim X_s$, and suppose that the dimension of the set of points of $X_s$ where $X_s$ is not separable over $k(s)$ is $<n$ (for example, $X_s$ separable). Then there exists an open neighbourhood $U$ of $s$ and an étale finite surjective morphism $S^{\prime \prime }\to U$ such that $X{\times }_S{S}^{\prime \prime }$ admits a section over S''. If $k(s)$ is infinite, we may take for S'' a closed subscheme of X_U.²⁵

Proof. Assume to start with that $k(s)$ is infinite. We proceed by induction on $n$, the case $n=0$ being trivial: in that case there exists an open neighbourhood $U$ of $s$ such that $X|U$ itself is étale, finite, and surjective above $U$, as one sees by immediate cross-references. If $n>0$, we apply (5.7.3) for the "separable" property, which allows us to replace $X$ by a "hyperplane section" $Y$ having the same properties up to the fact that $n$ is replaced by $n-1$. If $k(s)$ is not assumed infinite, we begin by making an étale base change; the argument goes through.

Remark (5.7.5).

In particular, if $X$ is projective and separable over $S$, it admits locally over $S$ étale multisections. But we note that one can give examples with $X$ proper and smooth (but not projective) over $S$ where the same conclusion fails. Of course, the projective assumption cannot be weakened in general to an assumption of quasi-projectiveness, as one sees, for example, by taking $X$ étale and not finite over $S$.²⁶

V.5.8. Dimension of the set of exceptional hyperplanes

(5.8.1). In the previous subsections, and notably in §§V.5.2 and V.5.3, we have given statements asserting that the set of $\xi \in P^{\vee }$ such that $Y_{\xi }$ has a certain property $P$ is constructible and that it contains the generic point $\eta$; or, equivalently, that the set Z_P of $\xi \in P^{\vee }$ "exceptional for $P$" is constructible and rare — i.e. its closure is of codimension $\ge 1$. (N.B. We suppose $S=\mathrm{Spec}(k)$.)

In certain cases we can make this statement more precise by giving a better upper bound on this codimension, which is important for certain questions. For example, if we see that this codimension is $\ge 2$, it follows that a "sufficiently general" straight line $D$ of $P^{\vee }$ does not intersect Z_P, whence the existence (if $k$ is infinite) of "linear pencils" of hyperplane sections $Y_{\xi }$ ($\xi$ a geometric point of $D$) all of which have the property $P$. (See the subsection on pencils of hyperplane sections for examples.²⁷)

From the point of view of presentation, since the results of the present subsection make some of the results of the previous subsections more precise, the question arises whether it is necessary to do this catching-up in a separate subsection, or to give a more precise version gradually as one moves along. Let the redactor decide.²⁸

(5.8.2). Let $Z$ be the set of $\xi \in P^{\vee }$ such that $\dim Y_{\xi }>\dim X-1$, and suppose that for every irreducible component $X_i$ of $X$ we have $\dim f(X_i)\ge 2$.²⁹ Then $Z$ is of codimension $\ge 2$ in $P^{\vee }$. This follows from (5.2.1) and (5.2.2) (which imply that every irreducible component of $Y$ dominates $P^{\vee }$) and from the dimension theory for the morphism $Y\to P^{\vee }$. Starting from this result we may give, as a corollary, the case where we start with a closed subset $T$ of $X$ and where we consider the dimension of the inverse images $T_{\xi }$ in the $Y_{\xi }$ ($\xi \in P^{\vee }$); we may even take for $Z$ the set of $\xi \in P^{\vee }$ such that there exists an irreducible component of $T_{k(\xi )}$ whose trace on $Y_{\xi }$ has excessive dimension. (N.B. We always assume that for every irreducible component $T_i$ of $T$ we have $\dim f(T_i)>0$.)

Finally, the most precise statement in this direction, and one that follows easily from the first formulation (for $X$ irreducible) and from (5.2.7), is the following modified statement: $F$ being coherent over $X$, suppose that for every prime cycle $T$ associated to $F$ we have $\dim f(T)>0$. Then the set of $\xi \in P^{\vee }$ such that ${\varphi }_{\xi }$ is not $F_{k(\xi )}$-regular is (constructible and) of codimension $\ge 2$. (The notation for ${\varphi }_{\xi }$ is that of §V.5.5.) We can give this as the principal assertion, and announce the previous assertions as corollaries, the proof being via one of the corollaries.

Note that with the preceding notation, if $\xi \in P^{\vee }-Z$, then for every $y\in Y_{\xi }$ we have $copro{f}_y{G}_{k(\xi )}=copro{f}_y{G}_{\xi }$, and consequently if $coprofF\le n$, then for $\xi \in P^{\vee }-Z$ we have $coprof{G}_{\xi }\le n$. In particular, if $F$ is Cohen-Macaulay, then for $\xi \in P^{\vee }-Z$, $G_{\xi }$ is Cohen-Macaulay. Finally, if $F$ is $(S_k)$, then $G_{\xi }$ is $(S_{k-1})$ for $\xi \in P^{\vee }-Z$ (see §0_IV).

(5.8.3). We note that if $F$ is $(S_k)$, it can happen, for some $\xi \in P^{\vee }$ such that ${\varphi }_{\xi }$ is $F_{k(\xi )}$-regular, that $G_{\xi }$ has a component of codimension $\ge 2$ failing $(S_k)$,³⁰ even if $F={\mathcal{O}}_X$, $k=1$, $X$ being geometrically integral of dimension two (resp. $k=2$, $X$ being geometrically integral and geometrically normal of dimension three). It is enough to start from a projective integral surface

$$X\subset {\mathbb{P}}^r$$

over $k$ algebraically closed having a point $x$ where $X$ is not Cohen-Macaulay; then for every hyperplane passing through $x$, the corresponding hyperplane section $Y_{\xi }$ admits $x$ as an associated embedded cycle. (Respectively, we start from a normal — hence (S_2) — integral variety $X\subset {\mathbb{P}}^r$ of dimension three having a point $x\in X$ where $X$ is not Cohen-Macaulay; then the $Y_{\xi }$ passing through $x$ are not CM, i.e. they fail (S_2) at $x$.)

In these examples the set of "exceptional" $\xi$ for the property $(S_k)$ contains the hyperplane of $P^{\vee }$ defined by $x\in P$ and is of codimension one (and not of codimension $\ge 2$). Compare (5.8.5) below for a general precise result along these lines.

Proposition (5.8.4).

Let $T$ be a closed subset of $X$, and suppose that $codim(T,X)\ge k$. Then for every $\xi \in P^{\vee }$ we have $codim(T_{\xi },Y_{\xi })\ge k-1$. Let $Z$ be the set of $\xi \in P^{\vee }$ such that $codim(T_{\xi },Y_{\xi })=k-1$ (i.e. $codim(T_{\xi },Y_{\xi })<k$). Then $Z$ is a constructible, nowhere dense subset of $P^{\vee }$, i.e. $\overline{Z}$ is of codimension $\ge 1$ in $P^{\vee }$.

In order for it to be of codimension $\ge 2$, it is necessary and sufficient that for every irreducible component $T_i$ of $T$ of codimension equal to $k$ and such that $\dim f(T_i)=0$, there should exist an irreducible component $X_j$

of $X$ such that $codim(T_i,X_j)=k$ and $\dim f(X_j)>0$ — i.e., if $f$ is quasi-finite and $k>0$, that $T$ does not have isolated points such that ${\dim }_{x}X=k$.

The first assertion follows immediately from the following lemma (5.8.4.1) (a), which is a remorseful afterthought to §V.5.5.

Lemma (5.8.4.1).

Let $X$ be a locally noetherian prescheme, let $L$ be an invertible module over $X$, $\varphi$ a section of $L$, $Y=V(\varphi )$, and $T$ a closed subset of $X$. Assume that $codim(Y,X)\ge k$. Then:

(a) $codim(T\cap Y,Y)\ge k-1$.

(b) In order to have $codim(T\cap Y,Y)=k-1$ (i.e. $codim(T\cap Y,Y)<k$), it is necessary and sufficient that there should exist an irreducible component $T_i$ of $T$ contained in $Y$ such that $codim(T_i,X)=k$ and such that for every irreducible component $X_j$ of $X$ containing $T_i$ with

  dim 𝒪_{X_j, T_i} = dim 𝒪_{X, T_i}  ( = k),

we have $X_j\not\subset Y$.

The verification of this lemma is immediate, given the general facts in §0_IV (Chapter IV) about dimension.

With the assumptions of (5.8.4), and by (5.8.4.1) (b), we see which are the exceptional hyperplanes $H_{\xi }$. If we exclude the set Z_0 of $\xi \in P^{\vee }$ such that there is an irreducible component $R$ of $T$ or of $X$ with $\dim f(R)>0$ and such that $R_{\xi }$ is of "dimension too large" (a set which is of codimension $\ge 2$ and in what follows does not count), the exceptional $H_{\xi }$ are those for which there exists a $T_i$ with $codim(T_i,X)=k$ and $\dim f(T_i)=0$, $f(T_i)\subset H_{\xi }$, and such that for every irreducible component $X_j\supset T_i$ of $X$ with $codim(T_i,X_j)=k$, we have $f(X_j)\not\subset H_{\xi }$. For a given $T_i$ with $codim(T_i,X)=k$, if there exists an $X_j$ with $codim(T_i,X_j)=k$ and such that $\dim f(X_j)=0$, then we will have $f(X_j)=f(T_i)\subset H_{\xi }$, and consequently $\xi$ would not be exceptional relative to the $T_i$. If, on the other hand, for every $X_j\supset T_i$ such that $codim(T_i,X_j)=k$ we have $\dim f(X_j)>0$, then for $\xi \in P^{\vee }-Z_0$, $\xi$ is exceptional relative to $T_i$ if and only if $f(T_i)\subset H_{\xi }$; the set of such $\xi$ is (the trace on $P^{\vee }-Z_0$ of) a hyperplane of $P^{\vee }$. This proves (5.8.4), and also proves the more precise result that the exceptional set is the union of a set of codimension $\ge 2$ and of a union of hyperplanes determined in the evident way by the above proof.

Grothendieck note. I am afraid that the writeup is quite floppy (or perhaps sloppy)³¹ since I have reasoned geometrically all the time without saying so, by taking points over an algebraically closed field. Of course, the condition announced in (5.8.4) is indeed geometric, so that we may suppose $k$ algebraically closed and argue for $k$-rational points.

Using (5.8.4), (5.7.4), and the end of (5.8.2), we obtain:

Corollary (5.8.5).

Suppose that for all associated prime cycles $R$ we have at most simply [some condition]³² and that $F$ satisfies $(S_k)$. In order that the (constructible) set of points of $P^{\vee }$ such that ${\varphi }_{\xi }$ is $F_{k(\xi )}$-regular and $G_{\xi }$ is $(S_k)$ should have a complement of codimension at least two, it is necessary and sufficient that the following hold: for every integer $n\ge 0$, denote by $Z_n$ the set of $x\in T=suppF$ such that $copro{f}_{x}F\ge n$;³³ then for every irreducible component $Z_{n,i}$ of $Z_n$ with $codim(Z_{n,i},T)=n+k+1$ and $\dim f(Z_{n,i})=0$, there exists an irreducible component $T_j$ of $T$ containing $Z_{n,i}$ such that $codim(Z_{n,i},T_j)=n+k+1$ and $\dim f(T_j)=0$.

When $f$ is quasi-finite, then for every closed subset $R$ of $X$,³⁴ we have $\dim f(R)=\dim R$, so that the criterion takes the following form: there does not exist an isolated point $z$ in any one of the $Z_n$ such that ${\dim }_{z}T(=\dim F_z)$ is equal to $n+k+1$.

When $F$ is equidimensional of dimension $d$, this condition is vacuous if $d\le k$ (and indeed we knew this, because in this case the hypothesis $(S_k)$ on $F$ is nothing else but the Cohen-Macaulay hypothesis), and if $d\ge k+1$ it means that the set $Z_{d-(k+1)}$ of points of $T$ where the codepth of $F$ is $>d-(k+1)$, i.e. the true depth of $F$ is $\ge k+1$ (even though, a priori, we only have true depth of $F$ $\ge k$ as a consequence of property $(S_k)$ and $k\le d$). If we no longer assume that $F$ is equidimensional, the desired condition may be expressed in the following simple way:

(5.8.6). For every closed point $x\in suppF$ such that $\dim F_x\ge k+1$, we have $prof{F}_x\ge k+1$.

The sufficiency is seen immediately by putting $Z=x$. The necessity is seen by noticing that for every $\xi$ such that ${\varphi }_{\xi }$ is $F_{k(\xi )}$-regular and $x\in Y_{\xi }$, we have

  dim G_{ξ, x} = dim F_x − 1,   prof G_{ξ, x} = prof F_x − 1,

so that $x$ fails by default the above condition: we have $prof{G}_{\xi ,x}\ge k$ but $\dim G_{\xi ,x}\ge k$, which shows that $G_{\xi }$ does not satisfy condition $(S_k)$ at $x$; but the set

of $\xi$ such that $x\in Y_{\xi }$ is of codimension one. (N.B. I implicitly assumed that $k$ is algebraically closed, the case to which we reduce immediately.) The preceding general criterion should be evident in the case of (5.8.6).

We now study the points $y$ of $Y$ that are not smooth for $Y_{\xi }$ relative to $k(\xi )$. We restrict ourselves to the case where $f:X\to P$ is unramified (practically, it will be an immersion) and where $X\to S$ is smooth. We do not necessarily assume that $S$ is the spectrum of a field.

Since $f$ is unramified, the canonical homomorphism $f^{\ast }({\Omega }_{P/S}^1)\to {\Omega }_{X/S}^1$ is surjective and its kernel is a locally free module over $X$, which we denote ${\nu }_{X/P}^{\vee }$; when $f$ is an immersion, this is nothing else but the conormal module $J/J^2$ defined by the ideal $J$ of $X$ in $P$, and we call it in every case the conormal module. Thus we have the exact sequence

  (a)        0 ⟶ ν^∨_{X/P} ⟶ f^*(Ω^1_{P/S}) ⟶ Ω^1_{X/S} ⟶ 0.

Let us observe that we also have over $P$ an exact canonical sequence (which should appear as an example in §IV.16, for example)

  (b)        0 ⟶ Ω^1_{P/S}(1) ⟶ E_P ⟶ 𝒪_P(1) ⟶ 0

— i.e. ${\Omega }_{P/S}^1(1)$ is canonically isomorphic to the kernel of the canonical homomorphism $E_P(-1)\to {\mathcal{O}}_P$ deduced from $E_P\to {\mathcal{O}}_P(1)$. Applying $f^{\ast }$:

  (b₁)       0 ⟶ f^*(Ω^1_{P/S})(1) ⟶ E_X ⟶ 𝒪_X(1) ⟶ 0,

which gives an explicit description of $f^{\ast }({\Omega }_{P/S}^1)(1)$ over $X$ and allows therefore to identify ${\nu }_{X/P}^{\vee }(1)$ with a submodule locally a direct factor of E_X — or, dually, ${\nu }_{X/P}(-1)$ is canonically isomorphic to a quotient module of $E_X^{\vee }$. Consequently $\mathbb{P}({\nu }_{X/P}(-1))=\mathbb{P}({\nu }_{X/P})$ can be canonically embedded in $\mathbb{P}(E_X^{\vee })=X{\times }_S{P}^{\vee }=X_{P^{\vee }}$ as a projective subfibration over $X$, hence as a closed subscheme. The latter is necessarily contained in $Y$ (from the fact that ${\Omega }_{X/P}^1(1)$ is contained in the kernel of $E_X\to {\mathcal{O}}_X(1)$).