Éléments de Géométrie Algébrique — English

V.5.9. Change of projective embedding

For every integer $n>0$, let ${\mathbb{P}}^{(n)}=\mathbb{P}(Sy{m}^n(\mathcal{E}))$; we have an evident immersion $u_n:\mathbb{P}\to {\mathbb{P}}^{(n)}$, since $\mathcal{O}(n)$ is generated by its sections over every affine open of $S$ and since $p_{\ast }({\mathcal{O}}_{\mathbb{P}}(n))\cong Sy{m}^n(\mathcal{E})$,¹ where $p:\mathbb{P}\to S$ is the projection. If $f:X\to \mathbb{P}$ is an unramified morphism (resp. an immersion), the same is true of $u_n\circ f:X\to {\mathbb{P}}^{(n)}$. There is sometimes an advantage, in the study of $X$, to replacing $f$ by $u_n\circ f$ so as to avoid a very special behaviour of $f$ that is occasionally embarrassing in certain respects. An example of such a peculiarity is the one indicated (sic) in §V.5.8 (b), where $Y_{sing}\to {\mathbb{P}}^{\vee }$ has image of dimension $r-1$ but gives rise to an inseparable extension of fields. Another is the case of quadric surfaces in ${\mathbb{P}}^3$, where all the singular hyperplane sections are geometrically reducible — in spite of the fact that $X$ is geometrically irreducible.

Proposition (5.9.2).

Suppose $S=\mathrm{Spec}(k)$, $X$ smooth over $k$, and $f:X\to \mathbb{P}$ unramified. Let $n\ge 2$ and consider $f_n=u_n\circ f$. Then $f_n:X\to {\mathbb{P}}^{(n)}$ satisfies the equivalent conditions of §V.5.8.8; in particular, for $\xi \in {\mathbb{P}}^{(n)\vee }$ in the complement of a set of codimension $\ge 2$, the corresponding hyperplane section $Y_{\xi }$ is smooth or admits only a finite number of non-smooth points, which are geometrically ordinary singularities. If $f$ is an immersion, there is at most one such singular point, and it is rational over $k(\xi )$.

Grothendieck note. One would have to state §V.5.8.8 in a form that does not exclude the case where $f$ is not an immersion. The verification is essentially trivial under condition (ii bis) of §V.5.8.8. Without a doubt we should make explicit in §V.5.9.1 that the hyperplane sections of $X$ relative to $f_n$ are nothing other than the "sections" of $X$ by hypersurfaces of degree $n$ in place of hyperplanes.

Proposition (5.9.3).

Suppose that $X$ is geometrically irreducible and that $\dim f(X)\ge 2$; let $x\in X(k)$ and let $n\ge 2$.

(a) Consider the linear family of hyperplane sections of $X$ relative to $f_n=u_n\circ f$ that pass through $x$; its generic element defines a $Y_{\xi }^{(n)}$ which is geometrically irreducible.

(b) Let $L$ be a linear subvariety of $\mathbb{P}$ passing through $f(x)$ and not containing $f(X)$. Consider the linear family of hyperplane sections of $X$ relative to $f_n$ defined by the hyperplanes of ${\mathbb{P}}^{(n)}$ "tangent to $L$ at $x$" (that is, defined by the $n$-forms on $\mathbb{P}$ which vanish to order at least two at $x$ along $L$); its generic member is a $Y_{\xi }^{(n)}$ which is geometrically irreducible.

(c) Suppose that $X$ is smooth at $x$ and that $n\ge 3$, and that $f(X)$ is not contained in a plane defined over $k$. Consider the family of hyperplane sections $Y_{\xi }^{(n)}$ of $X$ relative to $f_n$ which are "tangent to $X$ at $x$". Then the generic member of the latter defines a $Y_{\xi }^{(n)}$ that is geometrically irreducible.

The proof is essentially trivial in terms of the criteria at the end of the previous section. Taking an affine model of $\mathbb{P}$ containing $f(X)$, we are reduced — for (a) — to finding three polynomials in the coordinates $T_1,T_2,\cdots ,T_r$ of degree $\le 2$, say $P$, $Q$, $R$, such that $Q(t)/P(t)$ and $R(t)/P(t)$ are algebraically independent over $k$ in $K$ (where $K$ is the function field of $X$ and $t=(t_1,\cdots ,t_r)$ is the system of elements of $K$ defined by the $T_i$). For (b) we require in addition that $P$, $Q$, $R$ vanish to order at least two on $L$, which we may furthermore suppose defined by the equations $T_1=\cdots =T_s=0$. Finally, for (c) the situation is the same except that $L$ is the image of the tangent space to $X$ at $x$, and we may, if necessary, take $P$, $Q$, $R$ of degree 3 — i.e. one notch higher. The hypothesis that $\dim f(X)\ge 2$ says that the transcendence degree of $K(t_1,t_2,\cdots ,t_r)$ over $k$ is $\ge 2$, i.e. we can find $t_1,t_2$ algebraically independent. In (a) we therefore take $P=T_1$, $Q=T_1^2$, $R=T_1{T}_2$. In (b) we proceed analogously, noting that we may choose $t_1$ so that T_1 vanishes on $L$, due to the fact that $f(x)\not\subset L^{\ast }$; this implies that there exists an index $i$ between 1 and $s$ such that $t_i\ne 0$, so that $t_i$ is not constant (since $t_i$ vanishes at $x$), and hence $t_i$ is not algebraic over $k$.² (We may assume $k$ algebraically closed.) Case (c) follows from (b) except when $f(X)$ is contained in the image $L$ (under $f$) of the tangent space to $X$ at $x$. If $\dim X=2$, this case is effectively exceptional: the trace of a quadric surface tangent to a plane on that plane is in general formed by two intersecting lines and is therefore not irreducible. But to treat that case, in the forms $P$, $Q$, $R$ displayed above we may evidently replace $X$ by $L$ itself, where the solution is trivial.

If $\dim L=2$, take $P=T_r^2$, $Q=T_r^3$, $R=T_r^2{T}_{r-1}$, and note that $Q/P=T_r$ and $R/P=T_{r-1}$ are linear forms independent on $L$, hence algebraically independent. If $\dim L\ge 3$, then $T_{r-2},T_{r-1},T_r$ are linearly independent on $L$ and we take

  P = T_r², Q = T_{r−1}², R = T_{r−2}².

Grothendieck note: combining with §V.5.8.18, we obtain a Corollary (5.9.4) — to be stated.

Finally we must combine the latter with Proposition (5.9.2) in order to find a recapitulating theorem in the "excellent case".

Theorem (5.9.5).³

If $X$ is smooth, proper, and geometrically integral over $k$, and $f:X\to \mathbb{P}$ is unramified with $X$ of dimension $\ge 2$, then (by considering the result of §V.5.8 when $f:X\to \mathbb{P}$ is an immersion) the variety of singular points of $Y_{\xi }$ …

Grothendieck note. Ask Grothendieck. The statement of (5.9.5) is left as a placeholder in the prenote; the surviving source breaks off mid-formulation. See translator footnote.

V.5.10. Pencils of hyperplane sections and fibrations of blown-up varieties

(5.10.1). Let $Z$ be the $P$-exceptional set in ${\mathbb{P}}^{\vee }$ relative to a constructible property $P$ such that $Z$ is a constructible subset of ${\mathbb{P}}^{\vee }$. Suppose $S=\mathrm{Spec}(k)$. We shall see (cf. §V.5.12, where we catch up with material which should without a doubt have appeared in earlier numbers) that in order to have $codim(Z,{\mathbb{P}}^{\vee })\ge 2$ it is necessary and sufficient that "every sufficiently general line" $L$ in ${\mathbb{P}}^{\vee }$ not meet $Z$; or also (or even) it suffices that there exist a single line $L$ in ${\mathbb{P}}^{\vee }$ not meeting $\overline{Z}$.⁴ If $k$ is infinite, it is necessary and sufficient that there exist a single line $L$ in ${\mathbb{P}}^{\vee }$ that does not meet $\overline{Z}$. We call the linear pencil of hyperplane sections of $X$ defined by a line $L$ in ${\mathbb{P}}^{\vee }$ the $L$-prescheme Y_L (definition valid for any $S$). The previous remarks together with the results of §§V.5.8 and V.5.9 then give criteria for the existence of such pencils having all their fibres $Y_{\xi }$ ($\xi \in L$) satisfying the property $P$, first of all in the case where $S$ is an infinite base field. Taking into account §V.5.8.2, if for every associated prime cycle on $X$ we have $\dim f(T)>0$, then we can (by taking the property $P^{\prime }=P+$ regularity condition for ${\varphi }_{\xi }$) require that the pencil Y_L be flat over $L$. In the case where $S$ is arbitrary, we can again — proceeding by the procedure of §V.5.7.1 — construct such a pencil over an open neighbourhood of a given point $s$ of $S$, in view of the fact that $k(s)$ is infinite and provided we know $Z$ to be closed (which is assured in various miscellaneous cases by the results of §V.5.5 and the assumption that $X\to S$ is proper).

To do it right it would be convenient, after a general explanation of this type,

to give recapitulating statements where we effectively apply the preceding results to a certain number of properties of this nature (and also encompassing module properties). As a minimum in this direction, we must give here the reformulation of (5.9.5) in terms of linear pencils — a fact constantly used in geometric applications.

(5.10.2). By polarity, to a line $L$ in ${\mathbb{P}}^{\vee }$ there corresponds a linear subvariety $L_0$ of codimension 2 in $\mathbb{P}$ ($S$ arbitrary). Set $T=X{\times }_{\mathbb{P}}{L}_0$. Another way to describe $T$ is as follows: $L$ is defined by a locally free quotient of rank 2 of ${\mathcal{E}}^{\vee }$, or what is the same by a submodule $F$ of $\mathcal{E}$, locally a direct factor, everywhere of rank two. Consider the composed homomorphism

$$F_X⟶{\mathcal{E}}_X⟶{\mathcal{O}}_X(1);$$

then $T$ is nothing other than the scheme of zeros of this composition, or, equivalently, it is defined by the ideal $\mathcal{J}$ which is the image of the corresponding homomorphism (obtained by twisting by ${\mathcal{O}}_X(-1)$)

$$F_X(-1)⟶{\mathcal{O}}_X.$$

Suppose that this homomorphism is regular, which means that, if one writes down locally a totally ordered basis of $F_X(-1)$, then its image in ${\mathcal{O}}_X$ forms an ${\mathcal{O}}_X$-regular sequence — a condition that does not depend on the basis chosen, and that can be stated intrinsically by saying that $F_X(-1)\otimes {\mathcal{O}}_X/\mathcal{J}\to \mathcal{J}/{\mathcal{J}}^2$ is an isomorphism and that $V(\mathcal{J})=T\to X$ is a regular immersion.⁵ We then have:

Theorem (5.10.2).

Under the above hypothesis, the linear pencil Y_L together with the canonical projection $Y_L\to X$ is $X$-isomorphic in a unique fashion to the blow-up of the prescheme $X$ with centre $T$.

To understand the meaning of this theorem, it is convenient to note at the outset that, if $S=\mathrm{Spec}(k)$, then for a "sufficiently general" line $L$ in ${\mathbb{P}}^{\vee }$ the regularity condition is satisfied (cf. the catching-up indicated in §V.5.12, namely for §V.5.5.3). In what follows, when constructing "good" linear pencils, as anticipated at the beginning of the present number, we could require that the pencil so constructed also satisfy the said condition (which is a condition of the same type as, but different from, the one consisting in requiring that for every $\xi \in L$ the section ${\varphi }_{\xi }$ be $X_{k(\xi )}$-regular). We should include the condition in question in the proposed recapitulating statements.

On the other hand, in practice (5.10.2) is used only in the situation of (5.9.5), which makes it desirable not to state the reformulation of (5.9.5) in terms of pencils until after (5.10.2),

so as to be able to include in the statement in question the isomorphism of the pencil with a blow-up as well (i.e. to give a description of the situation permitting a suitable reference). We thus obtain, for every projective smooth geometrically connected $X$ of dimension $\ge 2$ over an infinite field $k$, a way of finding a non-empty closed smooth subscheme of codimension two at every one of its points, such that the blown-up scheme admits a fibration over ${\mathbb{P}}^1$, with all fibres geometrically integral and all fibres smooth except for at most finitely many, the latter having at most one geometrically singular point, such a point being rational over $k$ and geometrically an ordinary singularity.

This explains the importance of a deep study (just begun at present) of such fibrations with singular fibres, which to some extent reduces the study of projective smooth varieties of dimension $d$ to that of one-parameter families of projective varieties of dimension $d-1$ that may have ordinary singularities.

Statement (5.10.2) is a more or less immediate consequence of the following, which is completely independent of the story of hyperplane sections and would be better placed in a separate section on regular immersions.⁶

Grothendieck note. Proposition (5.10.3) is crossed out [in the source]. Ask Grothendieck if that is his intention.

Proposition (5.10.3).

Let $X$ be a prescheme, $\mathcal{G}$ a quasi-coherent module on $X$, and $u:\mathcal{G}\to \mathcal{O}$ a homomorphism; set $\mathcal{J}=u(\mathcal{G})$, $T=V(\mathcal{J})$. Let $\tilde{X}$ be deduced from $X$ by blowing up $T$. Consider, on the other hand, $p=\mathbb{P}(\mathcal{G})$, the canonical homomorphism ${\mathcal{G}}_p\to \mathcal{O}(1)$ and its kernel $\mathcal{H}$ (so that we have the exact sequence $0\to \mathcal{H}\to {\mathcal{G}}_p\to \mathcal{O}(1)\to 0$), the homomorphism $u_p:{\mathcal{G}}_p\to \mathcal{O}$, and the quasi-coherent ideal $\mathcal{K}=u_p(\mathcal{H})\to \mathcal{O}$. Then $\tilde{X}$ is canonically isomorphic to a closed subscheme of $V(\mathcal{K})$. If $\mathcal{G}$ is locally free and $u$ is "regular", then the above isomorphism is an isomorphism of $\tilde{X}$ with $V(\mathcal{K})$ itself; in this case $\mathcal{H}$ is locally free over $p$, and $\mathcal{H}\to {\mathcal{O}}_p$ (whose prescheme of zeros is $\tilde{X}$) is also regular.

The first statement is almost trivial. The second is an exercise which presents no difficulty (I have not done it in detail, thinking that you can deduce it just as well as I can).

If in (5.10.2) we have $S=\mathrm{Spec}(k)$ and $X$ of dimension $\ge 1$, then the regularity assumption made is equivalent to $T=\varnothing$, so that $Y_L\to X$ is an isomorphism. We therefore find, by combining with (5.9.2):

Corollary (5.10.4) (of (5.10.2)).

*Let $X$ be a smooth geometrically connected curve in a projective space $\mathbb{P}$ over an infinite field $k$, and let $n\ge 2$. Then there exists a linear pencil of $n$-forms on $\mathbb{P}$ defining a morphism $X\to {\mathbb{P}}^1$ having the following property: the morphism

is generically étale of degree $d$, and for every geometric point $s$ of ${\mathbb{P}}^1$, $X_s$ is étale over the algebraically closed field $k^{\prime }=k(s)$, or else it is $k^{\prime }$-isomorphic to the sum of $d-2$ schemes $\mathrm{Spec}{k}^{\prime }$ and the scheme $I_{k^{\prime }}=\mathrm{Spec}{k}^{\prime }[t]/(t^2)$. In the language of the forefathers: there is at most one ramification point, and it is "quadratic".*

V.5.11. Grassmannians

Since we shall now use linear subvarieties of $\mathbb{P}$ not only of relative dimension 0 and $n-1$, it is clear that we shall need some notations concerning Grassmannians and some [sorites]⁷ (facts) of "elementary geometry" flavour concerning the constructions involving linear varieties; these should all come at the beginning of the paragraph. In addition, in practice one sometimes takes arbitrary linear sections and not only hyperplane sections, and it is proper to revisit, in this enlarged spirit, all the previous numbers.

Let $\mathcal{E}$ be a quasi-coherent module on the prescheme $S$, and let $n$ be an integer > 0. Consider the functor $(Sch{)}^{\circ }/S\to (Ens)$ defined by

  Grass_n(ℰ)(S′) = {locally free quotient modules of rank n of ℰ_{S′}}.

This functor is representable, and the prescheme over $S$ which represents it will also be denoted $Gras{s}_n(\mathcal{E})$. To prove representability, consider the natural homomorphism of functors

  Grass_n(ℰ) ⟶ Grass_1(Λ^n ℰ) = ℙ(Λ^n ℰ)

defined by associating with every locally free quotient $G$ of rank $n$ of ${\mathcal{E}}_{S^{\prime }}$ the locally free module of rank one ${\Lambda }^{n}G$, considered as a quotient of ${\Lambda }^n{\mathcal{E}}_{S^{\prime }}$. As in Séminaire Cartan,⁸ one proves that this morphism is "representable by a closed immersion", so that $Gras{s}_n(\mathcal{E})$ appears as a closed subscheme of $\mathbb{P}({\Lambda }^n\mathcal{E})$; in particular it is separated over $S$ and quasi-compact over $S$, and if $\mathcal{E}$ is of finite type, it is projective over $S$. If $\mathcal{E}$ is of finite presentation, then so is $Gras{s}_n(\mathcal{E})$: indeed, we may suppose $S$ affine, $S=\mathrm{Spec}(A)$, so that $\mathcal{E}$ comes from a module of finite type over a subring of finite type of $A$ — since the formation of $Gras{s}_n(\mathcal{E})$ is evidently compatible with base change over $S$.

Since $\mathcal{E}$ is locally free, $Gras{s}_n(\mathcal{E})$ is smooth over $S$ with geometrically connected fibres. This is a consequence of the more precise fact: if $\mathcal{E}$ is free of rank $r$, then $Gras{s}_n(\mathcal{E})$ may be recovered from $\left(\genfrac{}{}{0ex}{}{r}{n}\right)$ open subsets, each of which is $S$-isomorphic to affine space of relative dimension $n(r-n)$ over $S$. This decomposition corresponds to the choice, thanks to a basis of $\mathcal{E}$, of $\left(\genfrac{}{}{0ex}{}{r}{n}\right)$ decompositions of $\mathcal{E}$ by exact sequences

  (s)   0 → ℰ′ → ℰ → ℰ″ → 0

with ${\mathcal{E}}^{\prime }$ locally free of rank $n$. Such an exact sequence allows us to define a subfunctor $Gras{s}_n(s)$ of $Gras{s}_n(\mathcal{E})$ by restricting to quotients $G$ of ${\mathcal{E}}_{S^{\prime }}$, locally free of rank $n$, such that the composed homomorphism ${\mathcal{E}}_{S^{\prime }}^{\prime }\to {\mathcal{E}}_{S^{\prime }}\to G$ is surjective (and hence bijective). The inclusion $Gras{s}_n(s)\to Gras{s}_n(\mathcal{E})$ is representable by an open immersion, and $Gras{s}_n(s)$ is canonically isomorphic to the fibre bundle $V({\mathrm{Hom}}_{{\mathcal{O}}_S}({\mathcal{E}}^{\prime },{\mathcal{E}}^{\prime \prime }))$.⁹

As a consequence, for example, of this particular structure, we may mention that if $s\in S$, then (since $\mathcal{E}$ is locally free of finite rank) every point of $Gras{s}_n(\mathcal{E})$ with values in $k(s)$ lifts to a section over a neighbourhood of $s$. On the other hand, if $S=\mathrm{Spec}(k)$ with $k$ an infinite field, then every non-empty open subset of $Gras{s}_n(\mathcal{E})$ contains a $k$-rational point. A point of $Gras{s}_n(\mathcal{E})$ with values in $S$, i.e. a locally free quotient module $G$ of rank $n$ of $\mathcal{E}$, canonically defines a subscheme of $\mathbb{P}(\mathcal{E})$ — namely $\mathbb{P}(G)$. Such a subscheme (without imposing or specifying¹⁰ the rank of $G$) is called a linear subvariety of $\mathbb{P}(\mathcal{E})$ (relative to $S$, if there is a possibility of confusion). It is therefore a projective fibration of relative dimension $n-1$ if $n\ge 1$, and empty if $n=0$. We immediately verify that the section of $Gras{s}_n(\mathcal{E})$, i.e. $G$, is known once one knows the corresponding linear subvariety $\mathbb{P}(G)$ of $\mathbb{P}(\mathcal{E})$. In this manner the Grassmannian can be interpreted as representing the functor "linear subvarieties of relative dimension $n-1$ of ${\mathbb{P}}_{S^{\prime }}$" for $S^{\prime }$ variable, with $n\ge 1$. It is furthermore possible to give an intrinsic characterization of this functor, i.e. of the notion of linear subvariety of relative dimension $m$ of "projective degree one" at every $s\in S$; this characterization will be given in a later chapter and we shall not need it at all here.

Suppose again that $\mathcal{E}$ is locally free of rank $r$, and let ${\mathcal{E}}^{\vee }$ be its dual. Then by polarity we find a canonical isomorphism $Gras{s}_n(\mathcal{E})\cong Gras{s}_{r-n}({\mathcal{E}}^{\vee })$ assigning to a quotient $G$ of $\mathcal{E}$ the quotient ${\mathcal{E}}^{\vee }/G^{\vee }$ of ${\mathcal{E}}^{\vee }$. From the point of view of linear varieties, to a linear variety $L$ of relative dimension $m$ of $\mathbb{P}$ there corresponds the linear dual variety $L_0$ of relative dimension $(r-1)-1-m$ of ${\mathbb{P}}^{\vee }$, i.e. of relative codimension $m+1$ in ${\mathbb{P}}^{\vee }$ (N.B.: $r-1$ is here the relative common dimension of $\mathbb{P}$ and ${\mathbb{P}}^{\vee }$ over $S$), which we may visualize geometrically as follows. Let us first take $n=r-1$; we find an isomorphism $\mathbb{P}({\mathcal{E}}^{\vee })\cong Gras{s}_{r-1}(\mathcal{E})$ that allows us to identify the points of ${\mathbb{P}}^{\vee }$ with values in $S$ (let us say) with linear subvarieties of codimension 1 of $\mathbb{P}$ (called again hyperplanes of $\mathbb{P}$).

This says that $L_0$ consists of hyperplanes which contain the linear subvariety $L$ of $\mathbb{P}$ (by which, of course, we mean that the points of $L_0$ with values in $S^{\prime }$ are the hyperplanes in ${\mathbb{P}}_{S^{\prime }}$ containing $L_{S^{\prime }}$). This follows from the fact (which should have occurred at the same time as the fact that a linear subvariety $L$ of $\mathbb{P}$ determines a locally free quotient $G$ of $\mathcal{E}$) that if $G$ and $G^{\prime }$ are two locally free quotients of $\mathcal{E}$ (not

necessarily of the same rank), then $\mathbb{P}(G^{\prime })\subset \mathbb{P}(G)$ (as linear subvarieties of $\mathbb{P}(\mathcal{E})$) if $G^{\prime }$ is dominated by $G$ (and the inclusion $\mathbb{P}(G^{\prime })\to \mathbb{P}(G)$ is nothing other than the morphism deduced from $G\to G^{\prime }$).

Here is a minimum of the [sorites]¹¹ which we must have at our disposal. The complete list cannot in any case be fixed until the other numbers of the present paragraph¹² are written up.

It seems convenient¹³ to introduce also the functor

  Grass(ℰ)(S′) = {set of locally free quotient modules (of unspecified rank) of ℰ_{S′}};

then $Grass(\mathcal{E})$ is representable by ${⨆}_{n\ge 0}Gras{s}_n(\mathcal{E})$. The linear subvarieties of $\mathbb{P}(\mathcal{E})$ are then defined by sections of $Grass(\mathcal{E})$ over $S$ (the rank, i.e. the relative dimension, may vary if $S$ is not connected).¹⁴

V.5.12. Generalization of the previous results to linear sections

Complements to notations. If $\mathbb{P}=\mathbb{P}(\mathcal{E})$ with $\mathcal{E}$ any quasi-coherent module, we also set $Gras{s}^n(\mathbb{P})=Gras{s}_{n+1}(\mathcal{E})$, so that $Gras{s}^n(\mathbb{P})$ corresponds to linear subvarieties of dimension $n$ in $\mathbb{P}$; this is valid for $n\ge -1$ if we agree that $\dim =-1$ means empty. If $\mathcal{E}$ is locally free, it is advisable to introduce

  Grass_n(ℙ) = Grass^{n−1}(ℙ^∨) = Grass_n(ℰ^∨),

which corresponds to linear subvarieties of codimension $n$ in $\mathbb{P}$. If $\mathcal{E}$ is of rank $r+1$,¹⁵ so that $\mathbb{P}$ is of relative dimension $r$, we have a canonical isomorphism $Gras{s}^n(\mathbb{P})\cong Gras{s}_{r-n}(\mathbb{P})$. In what follows we fix $\mathcal{E}$ locally free of rank $r$, and we are interested in linear subvarieties of $\mathbb{P}$ of given dimension $m$, i.e. in $G{r}_m=Gras{s}_m(\mathbb{P})=Gras{s}_m({\mathcal{E}}^{\vee })$.

Over this prescheme we have a canonical quotient $G$ locally free of rank $m$ of ${\mathcal{E}}_{Gr}$, which we denote $F$. The natural incidence prescheme over $\mathbb{P}\times G{r}_m$, which represents the subfunctor of the product functor corresponding to couples consisting of a section of ${\mathbb{P}}_{S^{\prime }}$ and a linear subvariety of codimension $m$ of ${\mathbb{P}}_{S^{\prime }}$ containing the former, can be made explicit as follows. Let $T=\mathbb{P}\times G{r}_m$ (or, if we prefer, any prescheme $T$ over this product); over $T$ we have ${\mathcal{E}}_T$, the quotient ${\mathcal{O}}_T(1)$, and the submodule $G_T^{\vee }$, locally a direct factor of ${\mathcal{E}}_T^{\vee }$. We consider the composition of the canonical homomorphisms $G_T^{\vee }\to {\mathcal{E}}_T^{\vee }\to {\mathcal{O}}_T(1)$ (which by transposition corresponds also to a homomorphism of the submodule ${\mathcal{O}}_T(-1)$ of ${\mathcal{E}}_T^{\vee }$ into the quotient $G_T^{\vee }$, namely ${\mathcal{O}}_T(-1)\to G_T$); this may also be

considered as defined by a section ${\varphi }_m\in \Gamma (T,G_T(1))$. The incidence prescheme (resp. its inverse image in $T$) is nothing other than the prescheme of zeros of one or the other of these homomorphisms, equivalently of the section ${\varphi }_m$. We could denote the incidence prescheme by $H^{(m)}$; for $m=1$ we recover the one from §V.5.1. If $X$ is over $\mathbb{P}$, we may set $Y^{(m)}=X{\times }_{\mathbb{P}}{H}^{(m)}$ and define by this the notation $Y_{\xi }^{(m)}$ if $\xi$ is a point of $G{r}_m$ with values in some $S^{\prime }$. Therefore the $Y_{\xi }^{(m)}$ are "linear sections" of $X$ over $\mathbb{P}$ (or rather of $X_{S^{\prime }}$ over ${\mathbb{P}}_{S^{\prime }}$) by linear subvarieties of codimension $m$ of $\mathbb{P}$ (or rather of ${\mathbb{P}}_{S^{\prime }}$).

Grothendieck note. I take this opportunity for a notational self-criticism that could come in §V.5.1: the present point arbitrarily assigns the letter $Y$ to an object $Y$ corresponding to $X$ (so that if $X$ becomes $Z$, we no longer know quite what to take). This inconvenience has already led me into some incoherent notations. Perhaps the more general context with an integer $m$ as here suggests a reasonable solution: to write $X^{(m)}$ in place of $Y^{(m)}$, hence $X^{(1)}$ in place of $Y$ in §V.5.1. In such a way we might approximately have $X^{(m)(m_1)}=>X^{m+m_1}$. I am going to try such notation in what follows. Evidently even the exponent is open to criticism, since it is current practice in algebraic geometry to denote by an exponent the dimension of the varieties which enter into play. But since we shall never make use of this type of convention, I think we have a free hand on that score.

We see immediately, in the preceding construction of $X^{(m)}$, that there is a canonical isomorphism $X^{(m)}=Gras{s}_m(F^{\vee })$, where $F_X\cong f^{\ast }({\Omega }_{\mathbb{P}/S}^1)(1)$ is the kernel of ${\mathcal{E}}_X\to {\mathcal{O}}_X(1)$; in particular $X^{(m)}$ is smooth over $X$ with geometrically integral fibres — in fact, rational varieties of dimension $m(r-m)$. Of course, the verification reduces to the case $X=\mathbb{P}$, and because of this it belongs — just like the previous considerations — to the generalities about Grassmannians (which I am sure you are going to develop in a separate paragraph).

We now have a perfect analogy with the diagram from §V.5.1. Again a forgotten point: as a prescheme over $G{r}_m$, $H^{(m)}$ is canonically isomorphic to $\mathbb{P}({\mathcal{E}}_{Gr}/G^{\vee })$; it is therefore an excellent projective fibration (though of course we may not conclude this in general for $X^{(m)}$ over $G{r}_m$).

Proposition (5.2.1) [transposes]¹⁶ without change. In (5.2.2) it should read: it is necessary and sufficient that for every $x\in Z$ we have $\dim x\le m-1$. For the proof we may, for example, reduce to (5.2.6) by considering a generic linear variety of codimension $m$ as the intersection of $m$ independent generic hyperplanes. Dieudonné se démerde [Latin/French].¹⁷

From the writing-up point of view, if (as seems preferable to me) we make $m$ general from the start, it seems preferable to prove (5.2.6) at the same time, where, of course, $\dim X-1$

is replaced by $\dim X-m$ (and where, by implication, dimension < 0 in the formula means that the considered set is empty).

Corollary (5.2.3) is read by replacing "finite" by "of dimension $\le m-1$". Corollary (5.2.4) is similar. The same for (5.2.5), replacing $\dim f(X_i)>0$ by $\dim f(X_i)\ge m$, and the same change in (5.2.7).

Proposition (5.2.8) remains true as stated. In (5.2.9) replace "finite" by "$\dim \le m-1$". The same for (5.2.10), (5.2.11).

Statement (5.2.12) remains valid as such, with a proof essentially unchanged (compare also the further comments below on §V.5.8); (5.2.13) replace "finite" by "$\dim \le m-1$". (5.2.14) stays valid as such; (5.2.15) by replacing "finite" by "$\dim \le m-1$". (5.2.16) is valid as stated. In (5.2.17) replace "finite" by "$\dim \le m-1$". (5.2.18) as it stands.

For (5.3.1), we can state it for any $m$, supposing that $\dim f(X)\ge m+1$; but I propose to keep the principal statement in the case of a hypersurface and to give the general case as a corollary or a remark (it can be deduced immediately by the usual procedure of taking independent generic hyperplanes).

At the very least, it would be amusing to make explicit the generalized version of Lemma (5.3.1.1). For (5.3.2) read $\dim f(X)\ge m+1$; in (5.3.3) replace $\dim f(X_i)\ge 2$ by $\dim f(X_i)\ge m+1$, and in the definition of $G$ replace $\dim f(Z)=0$ by $\dim f(Z)\le m-1$.

The general considerations of §V.5.4 apply as such to the case of any $m$. The same is true of (5.4.2) and (5.4.3), by replacing in (b)(v) and (vi) the dimension condition by $\dim f(X)\ge m-1$ or $m+1$.¹⁸ Analogous change in (5.4.4)(b).

The discourse¹⁹ of (5.5.1) goes through as such. In (5.5.2) it is necessary to remember that $\varphi$ becomes a section ${\varphi }^{(m)}$ of $G_T(1)$ (where $T=X_S\times G{r}_m$), inducing the sections ${\varphi }_{\xi }^{(m)}$ of ${\mathcal{O}}_{X_{k(\xi )}}(1{)}_{k(\xi )}\otimes G(\xi )$ for $\xi \in G{r}_m$.

But in general we shall explain in §V.5.19 that if we have a section $\varphi$ of a locally free module of rank $n$ over a prescheme, the assertion that such a section is $F$-regular for a given module $F$ means, in terms of a local basis, that we have an $F$-regular sequence of $m$ sections of ${\mathcal{O}}_X$ (and it will be necessary to verify that this is independent of the chosen basis). In the case $m=1$ we have the intrinsic, evident interpretation mentioned in (5.5.2). With this language convention, (5.5.3) remains valid as such; the same is true of (5.5.4).

The first part of Remark (5.5.5) admits a generalization to the case of arbitrary $m$: if F_S satisfies $(S_m)$, then the regularity condition mentioned for ${\varphi }_{\xi }$ can be expressed in a purely dimensional manner.

The second part of Remark (5.5.5) is valid as such for any $m$. Theorem (5.5.6) extends as such; so does (5.5.7).

Proposition (5.6.1) (amendment for §V.5.12). Read $\dim f(X_i)\ge m+1$ and later $\dim f(Z)\le m-1$.

The general discourse²⁰ of (5.7.1) is valid as such in the case of any $m$. (5.7.2), (5.7.3) mutatis mutandis (pay attention in (5.7.2) to the notation $m$, confusing there). On the other hand, in the proof of (5.7.4) we no longer need to proceed step by step (closer and closer); we may take straight away a linear section of codimension $m=n$.

In (5.8.2) replace the condition $\dim Y_{\xi }>\dim X$ by $\dim Y_{\xi }>\dim X-m$,²¹ and the hypothesis $\dim f(X_i)>0$ by $\dim f(X_i)\ge m$. Analogous modification in the sequel to (5.8.2). Since (5.8.3) gives an example, there is no point in changing it, so we keep $m=1$.

I leave it as an exercise to you²² to find good statements for any $m$ corresponding to (5.8.4), (5.8.5), and (5.8.6). It is not necessary to do this exercise unless you feel like doing it.

I think that essentially all the developments of §V.5.8 except (5.8.6) can be adapted to the case of linear sections with arbitrary $m$. To do it en forme would without doubt be a rather long and tedious exercise. I have to admit that I do not know any applications depending in an essential manner on the analysis of this more general situation, so we are not really obliged to include these developments in these Éléments. On the other hand, experience proves that the act of writing up in this more general context often forces one to better unscrew ("dévisser") matters and fait mieux comprendre le fourbis (the whole shebang),²³ often sans beaucoup plus de mal. In addition, a certain number of syntax exercises in a properly geometric context like this one will do no harm, and of course it is not at all excluded that we will one day use it or need it, and we will be happy to find it. Still, I leave the whole decision on this subject up to you, and I restrict myself simply to summarizing the statements that we could perhaps give in this connection (à ce propos).

Let us again assume that $f:X\to \mathbb{P}$ is unramified and that $X$ is smooth over $S$ with components of dimension $\ge m$. Then $X^{(m)}=V_0$, and we distinguish therefore the subprescheme $X_{sing}^{(m)}=V_1$ of singular zeros of ${\varphi }_m$ relative to $Gras{s}_m$, which is also formed geometrically of pairs $(x,\xi )$ such that the linear variety $L_{\xi }$ cuts the tangent space to $X$ at $x$ excessively (considered as linear subvarieties of $\mathbb{P}$), i.e. such that the two spaces do not generate all of $\mathbb{P}$. Contrary to what happens for $m=1$, if $m$ is arbitrary, the morphism $X_{sing}^{(m)}\to X$ is not in general smooth, since the variety of $L$ passing through $x$²⁴ and cutting a given linear subvariety $T\ni x$ excessively is not in general smooth over $k$: this variety is only the closure of the smooth subvariety formed by $\xi$ such that the dimension of $T\cap L_{\xi }$ is just one more than the "normal" dimension $n-m$ (where $n=\dim T$, $m=codimL$).²⁵ Barring an error, the set

(contained in the relative supersingular set) $V^{\prime \prime }$ introduced in §V.5.16 (supplements) is nothing other than the set formed by the couples $(x,L_{\xi })$ such that the dimension of $T_x\cap L_{\xi }$ is $\ge n-m+2$, so that $V^{\prime }-V^{\prime \prime }$ is smooth over $S$ and, barring an error, it is exactly the same as the set of smooth points of $V^{\prime }$ over $X$. (The verification of this point requires a study of the filtration of the Grassmann scheme according to the dimensions of intersection with $L$ variable and $T$ fixed; barring an error, we find that the following notch of the filtration is formed exactly of the non-smooth points of the previous notch (*) [stratum?],²⁶ when we define the filtration not just set-theoretically but also scheme-theoretically, using the lemma from page 16 of the supplements to §V.5.16. This study would form one of the numbers of a "geometric" paragraph devoted to Grassmannians.)

If we also define $V^{(k)}$ as the subscheme of $X^{(m)}$ corresponding to $\dim T_x\cap L\ge n-m+k$, we find by an immediate calculation that

  dim Grass_m(ℙ) − dim V^{(k)} = (k − 1)(n − m) + k²

at least for the reasonable restrictions $k\le m$, $k\le r-m$, up to an error of calculation. (N.B.: this follows more generally from a calculation of the dimensions of the "cells" entering into the filtration of the Grassmannian alluded to above.)

For $k=2$, we find a difference of dimension $\ge 4$, so that the image of $V^{\prime \prime }$ in Grass²⁷ is of codimension $\ge 4$. Hence, if we are interested in what happens outside of subsets of Grass of codimension $\ge 2$, we may forget $V^{\prime \prime }$.

On the other hand, in $X\times Gras{s}_m-V^{\prime \prime }$ over $Gras{s}_m$, the situation is the one of the good case anticipated in the supplements to §V.5.16. Relative to the base scheme $S$: $V^{\prime }-V^{\prime \prime }$ is indeed smooth over $S$ (being such over $X$), of relative dimension equal to one less than that of $Gras{s}_m$ over $S$ (as we see by putting $k=1$ in the above formula). Thus the results of loc. cit. apply; in particular we find that the set of supersingular points of ${\varphi }_m$ relative to $Gras{s}_m$ is nothing other than $V^{\prime \prime }\cup V_2$, where V_2 is the subprescheme of ramification of $V^{\prime }-V^{\prime \prime }\to Gras{s}_m$. We may therefore say that outside of $V^{\prime \prime }$, the supersingular zeros result from the coalescence of at least two ordinary singular zeros (but we do not have to say this).

In this way we have essentially the equivalent of (5.8.7)(a) and (b). It should be possible to give an equivalent condition for (5.8.7)(c) by using the explicit description of the tangent bundle to $Gras{s}_m$ (analogous to the case $m=1$): it implies²⁸ that, for a geometric point of $V^{\prime }-V^{\prime \prime }$ unramified over $Gras{s}_m$, knowing its image in $Gras{s}_m$ implies knowing its image in $\mathbb{P}$, in view of the fact that the first image is a smooth point of the closed image of $V^{\prime }$ in $Gras{s}_m$ (we assume $S$ is the spectrum of a field). I could give a more precise statement upon request.

Once we grant this, we have the evident corollaries generalizing (5.8.8), (5.8.9), (5.8.11). Without a doubt it is also possible to state, in the case of arbitrary $m$, the other propositions of §V.5.8.

If this demands additional writing-up effort, we could give up this generalization, even if we include the previous differential developments.

The same is true of the results of §V.5.9.

As for §V.5.10, the situation studied there generalizes to the case of any $m$ in the following manner. We fix a linear subvariety $C$ of $\mathbb{P}$ of codimension $m+1$, and we consider the projective space $Q$ of linear subvarieties $L$ of $\mathbb{P}$ of codimension $m$ passing through $C$. Then $Q$ is a closed subscheme of $Gras{s}_m$; in particular we can construct $X_Q^{(m)}$, which we propose to study.

A first point, which must in any case figure in the text, is that $X_Q^{(m)}\to X$ is again birational at least if $C$ cuts $X$ "regularly"; and precisely, $X_Q^{(m)}$ is in this case canonically isomorphic to the prescheme deduced from $X$ by blowing up $X{\times }_{\mathbb{P}}C$: the proof of this fact is nothing other than (5.10.2), via (5.10.3). A second point, of some interest but which we do not absolutely have to include, consists in saying that if we choose $C$ "sufficiently general", then $X_Q^{(m)}\to Q$ has certain pleasant properties, the most classical being the following. Suppose $X$ is smooth over $S=\mathrm{Spec}(k)$, of dimension $n\ge m$, proper and geometrically irreducible. Then, for "sufficiently general" $C$, the set $T$ of $\xi \in Q$ for which $X^{(m)}$ is not smooth of dimension $n-m$ over $k(\xi )$ is geometrically irreducible over $k(\xi )$ and of codimension one in $Q$, and the set $T^{\prime }$ of $\xi \in T$ for which $X^{(m)}$ is "supersingular" at at least one point is rare (nowhere dense) in $T$. Finally, if $f:X\to \mathbb{P}$ is an immersion, then after enlarging $T^{\prime }$ slightly, for every $\xi \in T-T^{\prime }$ there is exactly one non-smooth point in $X_{\xi }^{(m)}$, and the latter is rational over $k(\xi )$. I forgot to specify in the statement that we assume $X\to \mathbb{P}$ unramified and that we have to initially replace $f$ by ${\varphi }_n\circ f$, $n\ge 2$ (where ${\varphi }_n$ is defined in (5.9.1)). The most natural way of proving this statement seems to be to use the subscheme $Z$ (denoted $T$ in (5.8.8)) of $Gras{s}_m$ such that $X^{(m)}$ is "singular": we see that, under the given conditions, it is geometrically irreducible of codimension one and that the subscheme Z_1 corresponding to $X^{(m)}$ supersingular is nowhere dense.

It remains, therefore, to prove a lemma of the following nature: let $Z$ be a closed subset of $Gras{s}_m$ of codimension $q$; then, defining $Q(C)$ in terms of $C$ as above, for every $C$ "sufficiently general", the intersection $Q(C)\cap Z$ is of codimension $\ge q$ in $Q(C)$; also, if $Z$ is geometrically irreducible,²⁹ so is $Q(C)\cap Z$ if $Z$ is "sufficiently general".

V.5.14. Conic projections

Grothendieck note. We have already used conic projections in different contexts, notably at the end of §V.5.8, in the formulation of (5.10.4), and elsewhere; and the "sorite" that follows should without a doubt come sooner — at the beginning of the paragraph and eventually in the auxiliary paragraph "Grassmannian".

Let $C=\mathbb{P}(F)$ be a linear subvariety of $\mathbb{P}(\mathcal{E})=\mathbb{P}$ of relative dimension $r-m-1$ over $S$, i.e. of codimension $m+1$ in $\mathbb{P}$, so that $F$ is a quotient of $\mathcal{E}$, locally free of rank $r-m$, with $F=\mathcal{E}/G$ where $G$ is locally free of rank $m+1$. We have defined in the algebraic way of Chapter II a morphism

  p_C : ℙ − C = ℙ(ℰ) − ℙ(ℰ/G) ⟶ ℙ(G),

which we shall interpret geometrically and which will be called (because of the description that follows) the conic projection with centre $C$. (N.B.: we assume $r-m-1$ lies between $-1$ and $r-1$, i.e. $m$ is between 0 and $r$, nothing more.) For this, let us begin by interpreting $\mathbb{P}(G)$ as a closed subscheme of $Gras{s}_m(\mathbb{P})=Gras{s}_{r-m+1}(\mathcal{E})$ via the obvious homomorphism of functors $\mathbb{P}(G)\to Gras{s}_{r-m+1}(\mathcal{E})$ obtained by considering, for every invertible quotient $G/G^{\prime }$ of $G$, the locally free module of rank $(r-m)+1$ $\mathcal{E}/G^{\prime }$ of $\mathcal{E}$ (and the same after every base change). The above homomorphism of functors is a homomorphism of functors, and since the first scheme is proper over $S$ and the second separated, it is a closed immersion. More generally, we may need to make explicit the closed immersion of the Grassmannians of $G$, i.e. of $\mathbb{P}(G)$ (in the sense of functors), into those of $\mathcal{E}$, i.e. of $\mathbb{P}(\mathcal{E})$. The image (in the sense of functors) of the morphism obtained is formed by linear subvarieties $L$ of the desired dimension of $\mathbb{P}$ that contain $C$. Let us denote by $Q(C)$ this image, in the case we are studying (i.e. for the dimensions specified above), and identifying $\mathbb{P}(G)$ with $Q(C)$, the conic projection morphism

  p_C : ℙ − C ⟶ Q(C) ⊂ Grass_m(ℙ)

is nothing other than the one which associates with every section of $\mathbb{P}-C$ the unique linear subvariety $L$ of $\mathbb{P}$ of codimension $m$ containing both $C$ and the given section (note, of course, that "containing a section" means that the section factors through $L$).

If now we have $f:X\to \mathbb{P}$, it makes sense to consider the composition

  X − f^{−1}(C) ⟶ ℙ − C ⟶ Q(C),

which we may call the conic projection of $X$ relative to $f$ and with centre $C$, denoted $p_C^X$ or simply $p_C$. We point out that it is not in general defined on all of $X$; precisely, it is so if and only if $f^{-1}(C)=\varnothing$, i.e. if $f(X)$ does not meet the centre of the projection $C$. We shall

give another interpretation of this morphism in terms of constructions used in previous numbers. For this, with the notations introduced elsewhere, consider

                   q
       X ⟵──── X^{(m)}_{Q(C)} = X^{(m)} ×_{Grass_m} Q(C)
                              │
                              │ p
                              ↓
                            Q(C).

Note on the other hand that $q$ induces an isomorphism

  q′ : q^{−1}(X − f^{−1}(C)) ⟶ X − f^{−1}(C)

and it is immediate that $p_C$ is nothing other than $p^{\prime }\circ q^{\prime -1}$, where $p^{\prime }$ is the restriction of $p$ to $q^{-1}(X-f^{-1}(C))$. We may therefore say, using $q^{\prime }$ to identify purely and simply, that $p_C$ is the restriction of the morphism $p$ to $X-f^{-1}(C)\subset X_{Q(C)}^{(m)}$. For that reason it is convenient to denote again by $p_C^X$ or $p_C$, and to call the above morphism the extended conic projection of $X$ relative to $f:X\to \mathbb{P}$ with centre $C$.³⁰ In this way the properties of the restricted conic projection are reduced to those of the extended conic projection, which has been systematically studied elsewhere or is supposed to have been studied³¹ (cf. §§V.5.10 and V.5.12).

The main question that arises, when $S=\mathrm{Spec}(k)$, is: what are the properties of the conic projection of $X$ if we take $C$ to be generic in $Gras{s}_{m+1}(\mathbb{P})$?³² This requires that we make a base change $k\to K(\eta )$, i.e. that $C$ is then a linear subvariety of $X_{k(\eta )}$. From the standard arguments that have already been repeated several times, this allows us to conclude analogous properties for the conic projections corresponding to the points of $Gras{s}_{m+1}(\mathbb{P})$ belonging to a non-empty open set of the said Grassmannian, and finally, since $k$ is infinite, we conclude the existence of a (in fact of infinitely many) $C$ defined over $k$, i.e. a linear subvariety of $\mathbb{P}$ itself (without changing the base field) giving rise to a conic projection having the properties in question. It is (will be) proper to group this type of general explanation with those of the same type given in §V.5.4 and §V.5.7, and which we have already used more or less implicitly, for example in §V.5.13. It is also proper, while we are at it, to examine the relative properties of a sheaf $F$ over $X$ upon taking its inverse image $F_{Q(C)}^{(m)}$ over $X_{Q(C)}^{(m)}$. It is necessary, moreover, in the precise situation described here to simplify the notation: I propose $\tilde{X}(C)$ and $\tilde{F}(C)$, or simply $\tilde{X}$ and $\tilde{F}$ if there is no possibility of confusion (warning: this $F$ is not the same as at the beginning of this section). Roughly speaking (grosso modo), if we assume that $f$ is an immersion, the properties of the generic conic projection

are very different according to whether we assume $\dim X\ge m$ or $\dim X\le m$ — say $\dim X<m$. In what follows we consider the $C_{\eta }\subset {\mathbb{P}}_{k(\eta )}$ corresponding to the generic point $\eta$ of $Gras{s}_{m+1}$, and we dispense with making explicit the interpretation of the results obtained in terms of "almost all the points …".

To start with, we have already noticed in (5.5.3) (a "catching-up" due to the general case in §V.5.12) that $C$ cuts $X$ regularly; more precisely and more generally, for every quasi-coherent $F$ on $X$, the section ${\varphi }^{(m+1)}$ of the locally free module of rank $m+1$ over $X_{k(\eta )}$ whose scheme of zeros is $C$, is $F$-regular. By (5.10.2) this implies for example that the morphism $\tilde{X}(C_{\eta })\to X_{k(\eta )}$ identifies $\tilde{X}(C_{\eta })$ with the prescheme deduced from $X_{k(\eta )}$ by blowing up $f^{-1}(C_{\eta })=X{\times }_{\mathbb{P}}{C}_{\eta }$. In the case where $\dim f(X)\le m$, we will also have $f^{-1}(C)=\varnothing$, and consequently $\tilde{X}(C_{\eta })\cong X_{k(\eta )}$ is an isomorphism (and indeed the restricted conic projection is then defined on all of $X$ a priori). The question then arises of the dimension of the fibres of $p_C:\tilde{X}(C_{\eta })\to Q(C_{\eta })$, and we find the flatness of this morphism. We find:

Proposition (5.14.1).

Suppose that $X$ is irreducible — more generally, that for every irreducible component $X_i$ of $X$ the fibre of $X_i$ at the point $f(x_i)$ ($x_i$ = generic point of $X_i$) has a dimension (independent of $i$), which is for example the case with $d=0$ if $f:X\to \mathbb{P}$ is quasi-finite. Then:

(a) If $\dim f(X)>m$, then the dimensions of the fibres of $p_C:\tilde{X}(C_{\eta })\to Q(C_{\eta })$ are all equal to $\dim X-m$.

(b) If $\dim X\le m$ and if the non-empty fibres of $X_i$ over $\mathbb{P}$ are of dimension $d$, then the fibres of $p_C$ are all of dimension $d$,³³ so $p_C$ is finite (resp. quasi-finite) if $f:X\to \mathbb{P}$ is finite (resp. quasi-finite).

In case (a) we have already seen (I hope) that for every point $\xi$ of $Gras{s}_m(\mathbb{P})$ the dimension of $X_{\xi }^{(m)}$ is at least equal to $\dim X-m$; this is so in particular if $\xi$ gives a point of $Q(C_{\eta })$. For the opposite inequality, note that (working over the field $k^{\prime }=k(\xi )$) since $C_{\eta }{\times }_k{k}^{\prime }\subset L_{\xi }$ is a hyperplane of $L_{\xi }$, the dimension of $X^{(m)}=X{\times }_{\mathbb{P}}{C}_{\eta }$ is $\ge \dim X-m$ (since the base change $k(\eta )\to k^{\prime }$ transforms the latter prescheme into (X ×_ℙ L_ξ) ×_{L_ξ} (C_{η,k′})); otherwise we would have, in the contrary case, $\dim X^{(m+1)}=\dim X-m-1$ by §V.5.2 (reviewed in §V.5.10). Case (b) is treated analogously: if we have $\dim X_{\mathbb{P}}\times L\ge d+1$, or what is the same, $\dim f_{k^{\prime }}(X_{k^{\prime }})\cap L\ge 1$, then we would have by the same argument as above that $X^{(m+1)}\ne \varnothing$, contrary to what we have remarked before (5.14.1).

Corollary (5.14.2).

*Suppose that $X$ has dimension $m$ and that $f:X\to \mathbb{P}$ is finite

(resp. quasi-finite); then the morphism $p_{C_{\eta }}:X_{k(\eta )}\to Q(C_{\eta })$ is finite surjective (resp. quasi-finite dominant).*

Indeed, this morphism is quasi-finite, and since $\dim X_{k(\eta )}=\dim Q(C_{\eta })$ it is dominant. If $f$ is finite, then $p_{C_{\eta }}$ is also finite, hence proper, hence surjective, since it is dominant.

Corollary (5.14.3).

Under the conditions of (5.14.1)(a), if $X$ is Cohen-Macaulay, then the morphism $p_C:\tilde{X}(C_{\eta })\to Q(C_{\eta })$ is a Cohen-Macaulay morphism, and a fortiori flat.

For the proof compare the remark above on page 21, before §V.5.5,³⁴ which gives a result that is stronger (including (5.14.3)?), taking into account that the $C_{\xi }^{(m)}$ for $\xi \in \varphi (\eta )$ are $F_{k(\eta )}$-regular.

This corollary must be modified, but for simplicity we may assume that $f$ is quasi-finite. If $F$ is a Cohen-Macaulay module over $X$ and if, for every irreducible component $Z$ of Supp F, we have $\dim Z\ge m$, then $\tilde{F}(C_{\eta })$ is Cohen-Macaulay and a fortiori flat relative to $\varphi (C_{\eta })$.

We note that we cannot replace, to obtain the same conclusion $p_C$ flat, the Cohen-Macaulay hypothesis on $X$ by a simple dimension hypothesis. Let us, for example, assume that $f$ is an immersion and that $X$ is irreducible of dimension $m$, so that $p_C$ is quasi-finite, and since $X_{k(\eta )}$ and $\varphi (C_{\eta })$ are irreducible of the same dimension and the second one is regular, $p_C$ cannot be flat unless $X_{k(\eta )}$ is Cohen-Macaulay.

More delicate are the differential properties of the conic projection, notably for $X$ smooth over $k$ and $f:X\to \mathbb{P}$ unramified, studied in §V.5.12. Let us recall that outside of a subset $Z$ of codimension 1 of $Q(C)$, the morphism $p_{C_{\eta }}$ over $\tilde{X}(C_{\eta })$ is smooth. A more detailed analysis summarized in §V.5.12 shows (or will show, if we do not do it) that if the dimensions of the components of $X$ are $\ge m$, then outside of a subset $Z^{\prime }\subset Z$ of $Q(C)$ of codimension $\ge 2$, the fibres $p_C^{-1}(\xi )=X_{\xi }^{(m)}$ can only have at worst ordinary singular points in the geometric sense, and indeed (if $f$ is an immersion and $X$ is geometrically irreducible) at most one such point, the latter being necessarily rational over $k(\xi )$ — these assertions all being valid at least if $k$ is of characteristic zero, or with the condition of replacing $f$ by ${\varphi }_n\circ f$ ($n\ge 2$) as in §V.5.9.

It is also appropriate to give the differential properties of $p_{C_{\eta }}$ in the case where $\dim X\le m$, and consequently $p_{C_{\eta }}$ is defined on all of $X_{k(\eta )}$. I restrict myself to indicating the following properties; the proof should be easy and is left to Dieudonné (or Blass).³⁵

Proposition (5.14.4).

Suppose that $f:X\to \mathbb{P}$ is unramified and that $\dim X\le m$. Let $T$ be a finite subscheme of $X$. Then:

*(a) If $f$ is an immersion, the restriction of $p_C$ to $T_{k(\eta )}$ is radicial, i.e. "geometrically

injective". If, in addition, $Y$ is a closed subset of $X$ of dimension $\le m-1$, then

  p_{C_η}^{−1}(p_{C_η}(Y_{k(η)})) ∩ T_{k(η)} = ∅.
```*

*(b) If `X` is smooth at the points of `T`, then `p_{C_η}` is unramified at all the points of `T_{k(η)}` (and at the
points of `p_{C_η}^{−1}(p_{C_η}(T_{k(η)}))`).*[^v-5p2-36]

**Proposition (5.14.5).**

<!-- label: V.5.14.5 -->

*Suppose that `dim X ≤ m − 1`, `f : X → ℙ` is an immersion, and finally `X` is separable over `k`. Let `Y_η` be the
scheme-theoretic image of `X_{k(η)}` in `Q(C_η)`. Then the induced morphism `p_{C_η} : X_{k(η)} → Y_η` is birational
and, for every point `x` of `X_{k(η)}` over a closed point of `X`, `p_{C_η}` is étale at `x` and at the points of
`p_{C_η}^{−1}(p_{C_η}(x))`.*

Note the following consequence:

**Corollary (5.14.6).**

<!-- label: V.5.14.6 -->

*Let `X` be a projective algebraic scheme, irreducible and separable of dimension `n`, over an infinite field `k`.
Then there exists a birational morphism of `X` onto a hypersurface in `ℙ^{n+1}`.*

We must avoid believing — even if `X` is a closed smooth geometrically irreducible subset of `ℙ` of dimension
`m − 1 = n` — that the conic projection `p_C` is necessarily an immersion. Indeed, if `k` is infinite, this would
imply that there exists a `C` rational over `k` having the same property, i.e. that `X` is isomorphic to a non-singular
hypersurface in `ℙ^{n+1}`. But already for `n = 1` (so `X` an algebraic projective curve, smooth and connected over an
algebraically closed field), it is easy to construct examples where `X` cannot be embedded (*ne peut s'immerger*) in a
`ℙ²`. Also, in (5.14.4) we must avoid confusing the given statement with the assertion (in general false) that `p_C`
is itself a monomorphism (preceding counterexample, if `X` is smooth of dimension `m`), or that `p_C` should be
unramified. For the latter point, to convince ourselves take `X` a closed smooth subscheme, irreducible and of
dimension `m` (over `k` algebraically closed), such that we have an `X → Q ≅ ℙ^m` unramified: it will be étale for
reasons of dimension, but we can prove (see Ch. VIII) that this implies `X ≅ ℙ^m` (`ℙ^m` being simply connected). The
intuitive geometric meaning of (5.14.4) is that the ramification set of `p_{C_η}` is "variable" over `k`; more
precisely, the ramification set of `p_{C_ξ}`, for a variable `ξ` in an open set of `Grass_{m+1}(\overline{k})`, varies
in `X(\overline{k})` and does not admit any "fixed point". Of course, to justify in the present section the passage
from `η` generic to neighbouring points of `Grass_{m+1}(ℙ)`, and also if needed to be able to assume responsibility
for the general considerations of §V.5.7.1, we have to consider the diagram

```text
       X ⟵───── \widetilde{X}(C)
       │              │
       ↓              ↓
       X ⟵───── Q(C)

obtained, with the help of the different $C\in Gras{s}_{m+1}(S)$, and more generally those obtained after a base change $T\to S$ for the points $\xi \in Gras{s}_{m+1}(\mathbb{P})$:

       X_T ⟵───── \widetilde{X}(C_ξ) = X_T(C_ξ)
        │              │
        ↓              ↓
        T  ⟵───── Q(C_ξ)

as deduced by the base change $\xi :T\to Gras{s}_{m+1}(\mathbb{P})=T$, from the universal diagram (relative to the canonical point of $Gras{s}_{m+1}$ in $T$):

       X_T̃ ⟵───── \widetilde{X}(C)
         │              │
         ↓              ↓
         T̃  ⟵───── Q(C)

where $C$ is the canonical linear subvariety of ${\mathbb{P}}_T$. Then the above $\tilde{X}(C_{\eta })\to Q(C_{\eta })$ is nothing other than the morphism of generic fibres for the $T$-morphism $\tilde{X}(C)\to Q(C)$ of the latter diagram, and every constructible property for the morphism of generic fibres implies the same property for neighbouring fibres. From the notational point of view, $Q$ should be considered (and even introduced) as the name of the natural morphism of functors

  Grass_{m+1}(ℙ) ⟶ {subschemes of Grass_m(ℙ)}.

V.5.15. Axiomatization of certain of the previous results

Grothendieck note. I think, on the whole, that the results of §§V.5.2 to V.5.8, which are mostly true under more general conditions than for the family of hyperplanes (or for hypersurfaces of given degree) in projective space, it seems proper to adopt an axiomatic point of view. I am not quite sure right now whether we can give a generalization in this sense of Bertini-Zariski (and therefore of the results of §§V.5.4 and V.5.6); I have written to the competent authorities³⁶ on this subject (Serre, Zariski) to ask whether they have knowledge of such an extension. I have, anyway, the impression that hypotheses of simple differential nature, of the type of those given above, should suffice to imply Bertini-Zariski. If the competent people cannot inform us in a satisfactory manner, we should try to clear the matter up by our own means.

We start from a commutative diagram of morphisms of finite presentation

       𝒫 ⟵────── 𝒫̃
       │              │
       ↓              ↓
       S ⟵────── G

(in the case of the principal application, $\mathcal{P}$ is a projective fibration, $G$ a deduced Grassmannian (grasmannienne — adjective!),³⁷ $\mathcal{P}$ the incidence prescheme. In the most important cases, the corresponding morphism $\mathcal{P}\to {\mathcal{P}}_S\times G$ should be a closed immersion, and we consider $G$ as a parameter scheme of a family of closed fibre-subpreschemes of $\mathcal{P}$ over $S$. More precisely, if $\xi \in G$, then ${\mathcal{P}}_{\xi }$ is a closed subprescheme of ${\mathcal{P}}_{s,k(\xi )}$, where $s$ is the point of $S$ below $\xi$. For most statements in this context we have, no doubt, $S=\mathrm{Spec}(k)$.) In the general case we may consider $G$ as a parameter scheme of a family of preschemes over the fibres of $\mathcal{P}$ over $S$, with $\xi$ corresponding to ${\mathcal{P}}_{\xi }$ over ${\mathcal{P}}_{s,k(\xi )}$. Of course, in place of taking for $\xi$ an absolute point of $G$, we may also take a point with values in an $S$-prescheme $T$, and we obtain then ${\tilde{\mathcal{P}}}_{\xi }\to {\mathcal{P}}_T$ (a $T$-morphism which is a closed immersion in the case presented above).

If $f:X\to \mathcal{P}$ is a morphism, we put $X=X{\times }_{\mathcal{P}}\mathcal{P}$, and we obtain a diagram of the same type as the preceding square:

       X ⟵────── X̃
       │              │
       ↓              ↓
       S ⟵────── G.

It is evident that all the questions studied in §§V.5.2 to V.5.8 retain meaning in the general context just enunciated, and there is good reason to disentangle³⁸ the axiomatic conditions that ensure the conclusions drawn in the above numbers.

We will assume that $\mathcal{P}$ and $G$ are flat over $S$, $G$ being with geometrically irreducible fibres (so as to be able to consider the generic points!) of dimension $N$; the morphism $\mathcal{P}\to \mathcal{P}$ is assumed to be smooth with geometrically irreducible fibres of dimension $N-m$. Therefore the morphism $X\to X$ has the same properties. All the properties mentioned³⁹ are stable under base change over $S$ and can in particular be applied to the fibres.

Let us assume initially $S=\mathrm{Spec}(k)$. Let $Z$ be a closed subset of $X$ of dimension $d$, so that its inverse image $Z$ in $X$ is a closed subset of dimension $d+(N-m)=N+d-m$. If $d<m$, then $Z$ is of dimension $<N$, so that $Z\to G$ cannot be dominant; therefore, if $\eta$ is a generic point of $G$, we have $Z_{\eta }=\varnothing$. Indeed, this reasoning shows even (by replacing $Z$ by $f(Z)$) that if $\dim f(X)<m$, then $Z_{\eta }=\varnothing$. We want a condition on (D) ensuring that if $\dim f(Z)\ge m$, then $Z_{\eta }\ne \varnothing$. It seems that this must form a primitive axiom in this situation (in the setting of (5.2.2) it would result from a global argument rather (quite) special): for every closed irreducible subset $Z$ of $\mathcal{P}$ of dimension $m$, $Z_{\eta }\ne \varnothing$.

Let us again take a closed subset $Z$ of $X$ such that $\dim f(Z)\ge m$; we see that $Z_{\eta }\to G$ is dominant and consequently $Z$ is of dimension equal to dim Z̃ − dim G = dim Z̃ − m. These properties allow us to develop in the present context the results corresponding to (5.2.1) and (5.2.11). There is a condition over⁴⁰ (insuring the validity of (5.2.12), i.e. that if $X$ is smooth then $X$ is also such if we assume $f:X\to \mathcal{P}$ unramified). We assume now that $\mathcal{P}$ is smooth over $k$, $\mathcal{P}\to {\mathcal{P}}_k\times G$ quasi-finite, and that the following condition is satisfied (where we assume $k$ algebraically closed): for every $x\in \mathcal{P}(k)$ and for every vector subspace $V$ of dimension $n\ge m$ of the tangent space $T_x(\mathcal{P})$ to $\mathcal{P}$ at $x$, we consider the set $E(x,V)$ of $\xi \in G(k)$ such that $\mathcal{P}$ has a point over $x$ not satisfying the following set of conditions: ${\mathcal{P}}_{\xi }$ is smooth at $z$, the tangent morphism of ${\mathcal{P}}_{\xi }\to \mathcal{P}$ at $z$ mapping $T_z(\mathcal{P})\to T_x(\mathcal{P})$ is injective (i.e. ${\mathcal{P}}_{\xi }\to \mathcal{P}$ unramified at $z$), and its image is "transversal" to $V$, i.e. its sum with $V$ is $T_x(\mathcal{P})$. Then $E(x,V)$ (which we know to be the trace of a constructible well-defined set of $G$ in $G(k)$) is of dimension $\le N-n-1$. Using⁴¹ this condition, an application of the Jacobian criterion and a dimension count shows that the closed subset $E$ of points $x$ of $X$ such that $X\to G$ is non-smooth at $x$, or $\mathcal{P}\to G$ is not smooth at $f(x)$, or $\mathcal{P}\to \mathcal{P}$ is ramified at $f(x)$, is of dimension $\le n+(N-n-1)=N-1$ ($X$ being smooth everywhere of dimension $n$). Therefore $\dim E<N=\dim G$, so that $E_{\eta }=\varnothing$ and a fortiori $X_{\eta }$ is smooth over $k(\eta )$, and the developments of §V.5.5 are evidently valid in this current context.

The passage in §V.5.4 from a generic section to a general section, and the developments of §V.5.5, are evidently valid in the present context (but at this point are tautologies, or a reformulation of §§V.5.8, V.5.9, V.5.12 which we hesitate to state in their form). Likewise the developments of §V.5.7.1 are valid in every case if $k$ is algebraically closed (and even if $k$ is simply infinite, if we assume $G$ rational over $k$), and the special cases (5.7.2), (5.7.3); as for the result (5.7.4), it is evidently an application of a special nature to the situation of hyperplane sections. As I have said, §§V.5.3 and V.5.6 are suspended pending the extension of Zariski's theorem.

It remains to extend also the results of §V.5.8 (reconsidered in §V.5.12), which take on a more pleasant appearance. I advise you to begin formulating these results in this context, trying to go as far as possible in this way. I have the impression that we should be able to recover at least what is not a direct consequence of (5.8.7)(c) (we could even attempt to abstract the axiomatic conditions that allow one to go through a variant of (5.8.7)). I limit myself to these recommendations, but I am ready to come back to them in more detail if you have special difficulties.