ESSENTIAL DIMENSION OF QUADRICS

NIKITA KARPENKO AND ALEXANDER MERKURJEV

To the memory of Oleg Izhboldin

Abstract. Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dim_{es}(X) of X, as defined by Oleg Izhboldin, is

dim_{es}(X) = dim(X) − i(X) + 1 ,

where i(X) is the first Witt index of X (i.e., the Witt index of X over its own function field).

Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F (X). Our main theorem states that dim_{es}(X) ≤ dim(Y) and that in the case dim_{es}(X) = dim(Y) the quadric X_{F (Y)} is isotropic.

Applying the main theorem to a projective quadric Y, we get a proof of Izhboldin’s conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F (X), then dim_{es}(X) ≤ dim_{es}(Y), and the equality holds if and only if X is isotropic over F (Y).

Let (V, φ) be a non-degenerate quadratic form over a field F of characteristic not 2 and let X = Q(φ) be the quadric hypersurface given by the equation φ(x) = 0 in the projective space ℙ(V). We say that the quadric X is anisotropic if φ is an anisotropic quadratic form. By Springer’s theorem, every closed point of an anisotropic quadric X has even degree. Is it possible to compress X rationally, i.e., to find a rational morphism X → Y to a variety Y of smaller dimension with all closed points of even degree?

The quadratic form φ is isotropic over the function field F (X), hence, by the general theory of quadratic forms, φ_{F (X)} is isomorphic to ψ⊥kℍ for some anisotropic quadratic form ψ and some k ≥ 1, where ℍ stays for the hyperbolic plane. The number k is called the first Witt index of φ (or X), and we denote it by i(φ) (or i(X)). Let V′ ⊂ V be a subspace of codimension i(X) − 1. Since V′ ⊗ F (X) intersects nontrivially a totally isotropic subspace of V ⊗ F (X), the anisotropic quadric X′ = Q(φ|_{V′}) becomes isotropic over F (X), i.e., X compresses to the subvariety X′ of dimension dim(X) − i(X) + 1. The latter integer is denoted dim_{es}(X) and called the essential dimension of X.

We prove in the paper (Theorem 3.1) that an anisotropic quadric X cannot be compressed to a variety Y of dimension smaller than dim_{es}(X) with all closed points of even degree. Moreover, if there is a rational morphism X → Y with dim(Y) = dim_{es}(X), then there is a rational morphism Y → X, i.e., X is isotropic over F (Y).

Date: 11 June 2002.

Key words and phrases. Quadratic forms, first Witt index, complete varieties, Chow groups, correspondences. 2000 Mathematical Subject Classifications: 11E04; 14C25.

The second author was supported in part by NSF Grant #0098111.

Applying the main theorem to a projective quadric Y, we get a proof of Izhboldin’s conjecture (Theorem 4.1) stated as follows: if an anisotropic quadric Y becomes isotropic over F (X), then dim_{es}(X) ≤ dim_{es}(Y), and the equality holds if and only if X is isotropic over F (Y).

A field in the paper is an arbitrary field of characteristic not 2 (the characteristic restriction is important only there where quadratic forms are involved). By scheme we mean a separated scheme of finite type over a field, and by variety an integral scheme. We write CH_{d}(Y) for the d-th Chow group which is the group of rational equivalence classes of dimension d algebraic cycles on the scheme Y.

1. First Witt index of generic subforms

We are going to determine the first Witt index of certain subforms of a given anisotropic quadratic form. These subforms are generic in a sense (living over certain purely transcendental extensions of the base field), at least their first Witt indices turn out to be the minimal possible ones. The construction of these subforms is borrowed from [5, proof of lemma 7.9] (where a different property of these subforms is studied).

We recall that the first Witt index of an anisotropic quadratic F -form φ coincides with the minimal positive Witt index of φ_{E}, when E runs over all field extension of F. In particular, i(φ) ≤ i(φ_{L}) for any extension L/F such that φ_{L} is anisotropic.

Proposition 1.1. Let φ be an anisotropic quadratic F -form, and let n be an integer such that 0 ≤ n ≤ dim φ − 2. There exists a purely transcendental field extension F̃/F and an n-codimensional subform ψ ⊂ φF̃ such that

i(ψ) = \begin{cases}i(φ) − n, if n < i(φ); \\ 1, if n ≥ i(φ). \end{cases}

Proof. It suffices to give a proof for n = 1. Let t be an indeterminate. We consider the quadratic F (t)-form φ_{F (t)}⊥ ⟨−t⟩ and construct F̃ as its function field. The field extension F̃/F is clearly a purely transcendental one. Moreover, the anisotropic form φF̃ represents t, therefore φF̃ ≃ ψ⊥ ⟨t⟩ for certain 1-codimensional subform ψ ⊂ φF̃.

We are going to determine the first Witt index of ψ. First of all we clearly have: i(ψ) ≥ i(φ) − 1 and i(ψ) ≥ 1. Moreover, we have the following isomorphisms of F̃(φ)-forms (we omit the subscript F̃(φ) in the formula):

ψ⊥ℍ ≃ ψ⊥ ⟨t⟩ ⊥ ⟨−t⟩ ≃ φ⊥ ⟨−t⟩ ≃ φ′⊥ ⟨−t⟩ ⊥iℍ

where i = i(φ) and φ′ is the anisotropic part of the form φ_{F (φ)}. Cancelling one copy of ℍ, we get ψ ≃ φ′⊥ ⟨−t⟩ ⊥(i − 1)ℍ over F̃(φ). Note that the F̃(φ)- form φ′⊥ ⟨−t⟩ is anisotropic because the field extension F̃(φ)/F (φ)(t) is purely transcendental (by the reason that the F (φ)(t)-form φ⊥ ⟨−t⟩ is isotropic). Therefore the Witt index of ψ_{F̃(φ)} is i − 1. If i − 1 is positive, then i(ψ) ≤ i − 1 and we are done in this case. Otherwise i(ψ) ≤ i(ψ_{F̃(φ)}) = i(φ′_{F̃(φ)}⊥ ⟨−t⟩). Since the form φ′ remains anisotropic over the function field of the latter form (which is a purely transcendental extension of F̃(φ)), the latter first Witt index is equal to 1. □

Remark 1.2. In the case i(φ) > 1, the first Witt index of every 1-codimensional subform is known to be i(φ) − 1. This is a result due to A. Vishik which we do not use in this paper.

2. Correspondences

Let X and Y be schemes over a field F. Suppose that X is equidimensional and set d = dim(X). A correspondence from X to Y, denoted α: X ⇝ Y, is an element α ∈ CH_{d}(X × Y). A correspondence α is called prime if α is represented by a prime (irreducible) cycle. Every correspondence is the sum of prime correspondences.

Let α: X ⇝ Y be a correspondence. Assume that X is a variety and Y is complete. The projection morphism p: X × Y → X is proper and hence the push-forward homomorphism

p_{∗}: CH_{d}(X × Y) → CH_{d}(X) = ℤ · [X]

is defined [1, § 1.4]. The number deg(α) ∈ ℤ such that p_{∗}(α) = deg(α) · [X] is called the degree of α. Clearly, deg(α + β) = deg(α) + deg(β) for every two correspondences α, β: X ⇝ Y.

A correspondence α: Spec F → Y is represented by a 0-cycle z on Y. We set deg(z) = deg(α). This coincides with the usual notion of degree for 0-cycles as defined in [1, def. 1.4].

The image of a correspondence α: X ⇝ Y under the pull-back homomorphism

CH_{d}(X × Y) → CH_{0}(Y_{F (X)})

with respect to the flat morphism Y_{F (X)} → X × Y is represented by a 0-cycle on Y_{F (X)}. The degree of this cycle is equal to deg(α) (see [8, lemma 1.4]).

Lemma 2.1. Let F̃/F be a purely transcendental field extension. Then

deg CH_{0}(Y) = deg CH_{0}(YF̃) .

Proof. It suffices to consider the case where F̃ is the function field of the affine line 𝔸^{1}. The statement follows from the fact that the restriction homomorphism CH_{∗}(Y) → CH_{∗}(Y_{F (𝔸^{1})}) is surjective (cf. [7, proof of prop. 3.12]) as the composite of the surjections

CH_{∗}(Y) → CH_{∗+1}(Y × 𝔸^{1}) and CH_{∗+1}(Y × 𝔸^{1}) → CH_{∗}(Y_{F (𝔸^{1})})

(for the surjectivity of the first map see [1, prop. 1.9]). □

A rational morphism X → Y defines a degree 1 prime correspondence X ⇝ Y as the closure of its graph. Moreover, there are natural bijections between the sets of:

0) rational morphisms X → Y;

1) degree 1 prime cycles on X × Y;

2) rational points of Y_{F (X)}. Similarly, the following two sets are naturally bijective for every r > 0:

1) degree r prime cycles on X × Y;

2) closed points of Y_{F (X)} of degree r. Let g : Y → Y′ be a morphism of complete schemes. The image β of a

correspondence α: X ⇝ Y under the push-forward homomorphism

(id_{X} × g)_{∗}: CH_{d}(X × Y) → CH_{d}(X × Y′)

is a correspondence from X to Y′. The following statement is a consequence of functoriality of the push-forward homomorphisms:

Lemma 2.2. deg(β) = deg(α). □

Let X′ ⊂ X be a closed subvariety such that the embedding i: X′ ↪ X is regular of codimension r [1, B.7.1]. Then the embedding i × id_{Y} : X′ × Y ↪ X × Y is also regular of codimension r, hence the pull-back homomorphism

(i × id_{Y})^{∗}: CH_{d}(X × Y) → CH_{d−r}(X′ × Y)

is defined [1, § 6]. The pull-back γ of the correspondence α is a correspondence from X′ to Y.

Lemma 2.3. deg(γ) = deg(α). □

Proof. The statement follows from the commutativity of the diagram [1, th. 6.2]:

CH_{d}(X × Y) ⟶^{(i×id_{Y} )^{∗}} CH_{d−r}(X′ × Y)
p_{∗}↓  ↓^{p′_{∗}}
CH_{d}(X) ⟶^{i^{∗}} CH_{d−r}(X′),

where p and p′ are the projections. □

Let α: X ⇝ Y be a correspondence between schemes of dimension d. We write α^{t} for the element in CH_{d}(Y ×X) corresponding to α under the exchange isomorphism X × Y ≃ Y × X. The correspondence α^{t}: Y ⇝ X is called the transpose of α.

3. Main theorem

In this section X is an anisotropic projective quadric over a field F. We recall that the essential dimension dim_{es}(X) of X is defined as the integer dim(X) − i(X) + 1.

Theorem 3.1. Let X be an anisotropic projective F -quadric and let Y be a complete F -variety with all closed points of even degree. Suppose Y has a closed point of odd degree over F (X). Then

(1) dim_{es}(X) ≤ dim(Y);

(2) if, moreover, dim_{es}(X) = dim(Y), then X is isotropic over F (Y).

Proof. A closed point of Y over F (X) of odd degree gives rise to a prime correspondence α: X ⇝ Y of odd degree. By Springer’s theorem, to prove the statement (2) it is sufficient to find an odd degree correspondence Y ⇝ X.

Assume first that i(X) = 1, so that dim_{es}(X) = dim(X) . We prove both statements simultaneously by induction on n = dim(X) + dim(Y).

If n = 0, i.e., X and Y are of dimension zero, we have X = Spec K and Y = Spec L, where K and L are field extensions of F with [K : F] = 2 and [L : F] even. Taking the push-forward to Spec F of the correspondence α we get the formula

[K : F] · deg(α) = [L : F] · deg(α^{t}).

Since deg(α) is odd, α^{t}: Y ⇝ X is a correspondence of odd degree.

Assume that n > 0 and let d be the dimension of X. We are going to prove (2), so that we have dim(Y) = d > 0. It is sufficient to show that deg(α^{t}) is odd. Assume that the degree of α^{t} is even. Let x ∈ X be a closed point of degree 2. Since the degree of the correspondence Y × x: Y ⇝ X is 2 and the degree of x × Y : X ⇝ Y is zero, we can modify α by an appropriate multiple of x × Y and therefore assume that deg(α) is odd and deg(α^{t}) = 0. Hence the degree of the pull-back of α^{t} on X_{F (Y)} is zero. By [7, prop. 2.6] or [9], the degree homomorphism

deg: CH_{0}(X_{F (Y)}) → ℤ

is injective. Therefore there is a nonempty open subset U ⊂ Y such that the restriction of α on X × U is trivial. Write Y′ for the reduced scheme X \ U, i: X ×Y′ → X ×Y and j: X ×U → X ×Y for the closed and open embeddings respectively. The sequence

CH_{d}(X × Y′) ⟶^{i_{∗}} CH_{d}(X × Y) ⟶^{j^{∗}} CH_{d}(X × U)

is exact [1, prop. 1.8]. Hence there exists α′ ∈ CH_{d}(X × Y′) such that i_{∗}(α′) = α. We can view α′ as a correspondence X ⇝ Y′. By Lemma 2.2, deg(α′) = deg(α), hence deg(α′) is odd. Since α′ is a sum of prime correspondences, we can find a prime correspondence β : X ⇝ Y′ of odd degree, i.e., Y′ has a closed point of odd degree over F (X). The class β is represented by a prime cycle, hence we may assume that Y′ is irreducible. Since dim Y′ < dim X, by induction hypothesis, we get a contradiction with the statement (1).

In order to prove (1) assume that dim(Y) < dim(X). Let Z ⊂ X × Y be a prime cycle representing α. Since deg(α) is odd, the field extension F (X) ↪ F (Z) is of odd degree. The restriction of the projection X × Y → Y gives a proper morphism Z → Y. Replacing Y by the image of this morphism, we come to the situation where Z → Y is a surjection and so, the function field F (Z) is a field extension of F (Y).

In view of Proposition 1.1, extending the scalars to a purely transcendental extension F̃ of F, we can find a subquadric X′ of X of the same dimension as Y having i(X′) = 1. We note that according to Lemma 2.1, the hypothesis on X and Y is still satisfied over F̃. By Lemma 2.3, the pull-back of α with respect to the regular embedding X′ × Y ↪ X × Y produces an odd degree correspondence X′ ⇝ Y. Since dim(X′) < dim(X), by the induction hypothesis, the statement (2) holds for X′ and Y, that is, there exists an odd degree (in fact, of degree 1) correspondence β: Y ⇝ X′. We compose β with the embedding X′ ↪ X to produce an odd degree (in fact, of the same degree as β) correspondence γ : Y ⇝ X (Lemma 2.2). We may assume that γ is prime. Let T ⊂ Y × X be a prime cycle representing γ. Since the degree of γ is odd, the projection T → Y is surjective, so that F (T) is a field extension of F (Y) of odd degree.

Using the odd degree prime correspondences α: X ⇝ Y and γ : Y ⇝ X, we are going to construct an odd degree correspondence δ: X ⇝ X with even deg(δ^{t}) getting this way a contradiction with

Theorem 3.2 ([8, th. 6.4]). Let X be an anisotropic quadric with i(X) = 1. Then for every correspondence δ on X ×X, one has deg(δ) ≡ deg(δ^{t}) (mod 2).

Note that in the case where Y is regular we can simply take δ as the composite of the correspondences α and γ (cf. [8, proof of prop. 7.1]).

Lemma 3.3. Let F ↪ L and F ↪ E be two field extensions with odd degree [L : F]. Then there is a field K and field extensions L ↪ K and E ↪ K such that [K : E] is odd.

Proof. We may assume that L is generated over F by one element, say θ. Let f ∈ F [t] be the minimal polynomial of θ (of odd degree). Choose an odd degree irreducible polynomial g ∈ E[t] dividing f and set K = E[t]/gE[t]. □

By Lemma 3.3 applied to the field extensions F (T) and F (Z) of F (Y), we can find a field extension K of F (T) and F (Z) such that [K : F (Z)] is odd. Let a variety S over F be a projective model of the field extension K/F. Replacing S by the closure of the graph of the rational morphism S → Z × T, we come to the situation where the rational morphisms S → Z and S → T are regular. Let f be the composite of S → Z with Z → X and g be the composite of S → T and T → X. We write δ for the correspondence X ⇝ X given by the image of the morphism (f, g): S → X × X. The degree

deg(δ) = [F (S) : F (X)] = [F (S) : F (Z)] · [F (Z) : F (X)]

is odd and the degree of the transpose of δ is zero since g is not surjective as dim T = dim Y < dim X, a contradiction.

We have proven Theorem 3.1 in the case i(X) = 1. Consider now the general case (the first Witt index of X is arbitrary). Let X′ be a subquadric of X with dim(X′) = dim_{es}(X′) = dim_{es}(X) which we may find after extending the scalars to a purely transcendental extension according to Proposition 1.1. By Lemma 2.3, the pull-back β: X′ ⇝ Y of α with respect to the embedding of X′ into X is an odd degree correspondence. Therefore dim X′ ≤ dim Y by the first part of the proof. If dim X′ = dim Y, then again by the first part of the proof, X′ and hence X have rational points over F (Y). □

Remark 3.4. For X and Y as in part (2) of Theorem 3.1, assume additionally that dim(X) = dim_{es}(X), i.e., i(X) = 1. In the proof of Theorem 3.1, it is shown that deg(α^{t}) is odd for every odd degree correspondence α: X ⇝ Y.

We have also the following more precise version of Theorem 3.1:

Corollary 3.5. Let X and Y be as in Theorem 3.1. Then there exists a closed subvariety Y′ ⊂ Y such that

(i) dim(Y′) = dim_{es}(X); (ii) Y′_{F (X)} possesses a closed point of odd degree; (iii) X_{F (Y′)} is isotropic.

Proof. Let X′ ⊂ X be a subquadric with dim(X′) = dim_{es}(X). Then, by Theorem 4.1, dim_{es}(X′) = dim(X′). An odd degree closed point on Y_{F (X)} gives an odd degree correspondence X ⇝ Y which in turn gives an odd degree correspondence X′ ⇝ Y. We may assume that the latter correspondence is prime and take a a prime cycle Z ⊂ X′×Y representing it. We define Y′ as the image of the proper morphism Z → Y. Clearly, dim(Y′) ≤ dim(Z) = dim(X′) = dim_{es}(X). On the other hand, Z gives an odd degree correspondence X′ ⇝ Y′, therefore dim(Y′) ≥ dim(X′) by Theorem 3.1, and condition (i) of Corollary 3.5 is satisfied. Moreover, Y′_{F (X′)} has a closed point of odd degree. Since the field F (X × X′) is purely transcendental over F (X′) as well as over F (X), Lemma 2.1 shows that there is an odd degree closed point on Y′_{F (X)}, that is, the condition (ii) of Corollary 3.5 is satisfied. Finally the quadric X′_{F (Y′)} is isotropic by Theorem 3.1; therefore X_{F (Y′)} is isotropic. □

Example 3.6. Let the dimension of X be equal to 2^{n} − 1 for some n. Since there exists a field extension E/F such that X_{E} is given by an anisotropic Pfister neighbor ([2, th. 2]), the first Witt index of X is 1 by Theorem 4.1, that is, dim_{es}(X) = dim(X). The first part of Theorem 3.1 is therefore a generalization (to the case of arbitrary dim(X) and of arbitrary complete Y) of [2, th. 1], while the second part generalizes [4, th. 0.2].

4. Application to the algebraic theory of quadratic forms

Now we apply Theorem 3.1 to a special (but may be the most interesting) case where the variety Y is also a projective quadric:

Theorem 4.1. Let X and Y be anisotropic quadrics over F and suppose that Y is isotropic over F (X). Then

(1) dim_{es}(X) ≤ dim_{es}(Y);

(2) moreover, the equality dim_{es}(X) = dim_{es}(Y) holds if and only if X is isotropic over F (Y).

Proof. Let us choose a subquadric Y′ ⊂ Y with dim(Y′) = dim_{es}(Y) (we can do it over a pure transcendental extension of the base field by Proposition 1.1). Since Y′ becomes isotropic over F (Y) and Y is isotropic over F (X), Y′ is isotropic over F (X). According to Theorem 3.1, dim_{es}(X) ≤ dim(Y′). Moreover, in the case of equality, X is isotropic over F (Y′) and hence over F (Y). Conversely, if X is isotropic over F (Y), interchanging the roles of X and Y, we get as above the inequality dim_{es}(Y) ≤ dim_{es}(X), hence the equality holds. □

Example 4.2. If some anisotropic 11-dimensional quadratic form is not a Pfister neighbor, then its first Witt index is 1 (this is a result due to B. Kahn with an elementary proof given in [3]). Therefore, we recover a theorem of O. Izhboldin ([6, th. 5.3]) stating that an anisotropic 10-dimensional quadratic form remains anisotropic over the function field of any quadratic form of dimension > 10, if this second form is not a 4-fold Pfister neighbor.

Example 4.3. Similarly, if some anisotropic 13-dimensional quadratic form is not a Pfister neighbor, then its first Witt index is 1. Therefore, an anisotropic 12-dimensional quadratic form remains anisotropic over the function field of any quadratic form of dimension > 12, if this second form is not a 4-fold Pfister neighbor. This result is new.

Example 4.4. Coming back to the 11-dimensional forms, we also see, that if some anisotropic 11-dimensional form ψ becomes isotropic over the function field of another 11-dimensional form φ and φ is not a Pfister neighbor, then φ is isotropic over the function field of ψ. In the situation where the Schur index of the even Clifford algebra of ψ is at least 16, it can be then shown that φ is similar to ψ. This result is of particular interest in view of attempts to construct a field with the u-invariant 11.

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Laboratoire des Mathématiques, Faculté des Sciences, Université d’Artois, Rue Jean Souvraz SP 18, 62307 Lens Cedex, France

Department of Mathematics, University of California, Los Angeles, CA 90095-1555

E-mail address: karpenko@euler.univ-artois.fr
E-mail address: merkurev@math.ucla.edu