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 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 is isotropic over F (Y). 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). We also solve Knebusch’s problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dim_{es}(X).

Let (V, φ) be a non-degenerate quadratic form of dimension at least 2 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 over F (X) 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 (Corollary 3.4) 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., the quadric X is isotropic over F (Y). Applying this result to a projective quadric Y, we get a proof of Izhboldin’s conjecture (Theorem 4.1) stated as follows (cf. [4]): 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).

Date: December 2002.

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

We work with correspondences rather than with rational morphisms. The main result of the paper (Theorem 3.1) is formulated in terms of correspondences that provide a more flexible tool for study of relations between varieties.

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 Chow group of rational equivalence classes of dimension d algebraic cycles on a scheme Y.

1. First Witt index of generic subforms

Let φ be a (non-degenerate) quadratic form over F. Recall that φ ≃ φ_{an}⊥kℍ for an anisotropic form φ_{an}. The integer k is the Witt index i_{W} (φ) of φ.

For a field extension L/F, φ_{L} denotes the form φ ⊗_{F} L. If dim φ ≥ 3 or dim φ = 2 and φ is anisotropic, we write F (φ) for the function field F (X) of the (integral) quadric X defined by φ. The first Witt index i(φ) of an anisotropic form φ is defined as i_{W} (φ_{F (φ)}) ≥ 1.

By a subform of a quadratic form φ we mean the restriction of φ on a linear subspace of the vector space of φ and by a subquadric of the quadric X defined by φ we mean the closed subscheme of X given by a subform of φ. For the reader’s convenience we collect some basic properties of the Witt indices in the following

Lemma 1.1. Let φ be an anisotropic quadratic form over F.

(1) The first Witt index i(φ) coincides with the minimal Witt index of φ_{E}, when E runs over all field extension of F such that the form φ_{E} is isotropic.

(2) For any field extension L/F such that φ_{L} is anisotropic, we have i(φ_{L}) ≥ i(φ).

(3) For an r-codimensional subform ψ of φ and every field extension E/F, one has i_{W} (ψ_{E}) ≥ i_{W} (φ_{E}) − r and therefore i(ψ) ≥ i(φ) − r.

Proof. The first statement is proven in [7] and the second follows from the first one. The intersection of a maximal isotropic subspace U (of dimension i_{W} (φ_{E})) of the form φ_{E} with the space of the subform ψ_{E} is of codimension at most r in U, whence the third statement. □

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 (defined 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 [3, proof of lemma 7.9] (where a different property of these subforms is studied).

Proposition 1.2. 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 of φF̃ such that

i(ψ) = \{ \begin{matrix}i(φ) − n, if i(φ) > n; \\ 1, if i(φ) ≤ n. \end{matrix}

Proof. It suffices to give a proof for n = 1 in which case dim φ ≥ 3. Let t be an indeterminate. We consider the quadratic F (t)-form η = φ_{F (t)}⊥ ⟨−t⟩ and let F̃ be the function field of η over F (t) . The field extension F̃/F is clearly purely transcendental. Moreover, the anisotropic form φF̃ represents t, therefore φF̃ ≃ ψ⊥ ⟨t⟩ for a certain 1-codimensional subform ψ of φF̃ over F̃.

We are going to determine the first Witt index of ψ. We set i = i(φ). First of all, by Lemma 1.1(3), we have i(ψ) ≥ i − 1. Let φ′ be the anisotropic part of the form φ_{F (φ)}. We write τ for the form φ′⊥ ⟨−t⟩ over F (φ)(t).

There are 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 = τ⊥iℍ.

Cancelling one copy of ℍ, we get ψ ≃ τ⊥(i − 1)ℍ over F̃(φ). Note that the form τ_{F̃(φ)} is anisotropic because the field extension F̃(φ)/F (φ)(t) is purely transcendental (by reason of the isotropy of the form η_{F (φ)(t)}). Therefore the Witt index of ψ_{F̃(φ)} is i − 1. If i − 1 is positive, then i(ψ) ≤ i − 1 by Lemma 1.1(1), and we are done with this case. Otherwise i = 1, ψ ≃ τ over F̃(φ) and by Lemma 1.1(2),

(1)  1 ≤ i(ψ) ≤ i(ψ_{F̃(φ)}) = i(τ_{F̃(φ)}).

The field extension F (φ)(t)(τ)/F (φ) is clearly purely transcendental. The form η is isotropic over F (φ)(t)(τ), hence the field F̃(φ)(τ) = F (φ)(t)(τ)(η) is purely transcendental over F (φ)(t)(τ) and therefore over F (φ). It follows that the form φ′ remains anisotropic over F̃(φ)(τ). The form φ′ is a subform of τ of codimension 1 over F (φ)(t), hence by Lemma 1.1(3),

1 ≤ i_{W} (τ_{F̃(φ)(τ)}) ≤ i_{W} (φ′_{F̃(φ)(τ)}) + 1 = 1

and therefore, i(τ_{F̃(φ)}) = 1 and i(ψ) = 1 by (1). □

Remark 1.3. In the case i(φ) > 1, the first Witt index of every 1-codimensional subform is known to be i(φ) − 1. This result is due to A. Vishik [11, cor. 3] which we do not use in this paper. It readily follows from Theorem 4.1.

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 (integral) cycle. Every correspondence is a linear combination of prime correspondences with integer coefficients.

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 [2, § 1.4]. The number mult(α) ∈ ℤ such that p_{∗}(α) = mult(α) · [X] is called the multiplicity of α. Clearly, mult(α + β) = mult(α) + mult(β) for any 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) = mult(α). This coincides with the usual notion of degree for 0-cycles as defined in [2, 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 mult(α) (see [6, 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. [5, 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 [2, prop. 1.9]). □

Let X and Y be varieties over F and dim(X) = d. The generic point of a multiplicity r > 0 prime d-dimensional cycle Z ⊂ X × Y defines a degree r closed point of the generic fiber Y_{F (X)} of the projection X × Y → X and vice versa. Hence the following two sets are naturally bijective for every r > 0:

1) multiplicity r prime d-dimensional cycles on X × Y;

2) closed points of Y_{F (X)} of degree r. A rational morphism X → Y defines a multiplicity 1 prime correspondence

X ⇝ Y as the closure of its graph. Conversely, a multiplicity 1 prime cycle Z ⊂ X × Y is birational to X and therefore the projection to Y defines a rational map X ∼ Z → Y. Hence there are natural bijections between the sets of:

0) rational morphisms X → Y;

1) multiplicity 1 prime d-dimensional cycles on X × Y;

2) rational points of Y_{F (X)}. A multiplicity r prime correspondence X ⇝ Y can be viewed as a “generi-

cally r-valued map” between X and Y.

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. mult(β) = mult(α). □

Let X′ ⊂ X be a closed subvariety such that the embedding i: X′ ↪ X is regular of codimension r [2, B.7.1]. Then for every scheme Y, 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 [2, § 6]. The pull-back γ of a correspondence α : X ⇝ Y is a correspondence from X′ to Y.

Lemma 2.3. mult(γ) = mult(α). □

Proof. The statement follows from the commutativity of the diagram [2, 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 varieties 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 multiplicity. By Springer’s theorem [8, ch. VII, th. 2.3], to prove the statement (2) it is sufficient to find a closed point of X_{F (Y)} of odd degree or equivalently, to find an odd multiplicity 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] · mult(α) = [L : F] · mult(α^{t}).

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

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 mult(α^{t}) is odd. Assume that the multiplicity of α^{t} is even. Let x ∈ X be a closed point of degree 2. Since the multiplicity of the correspondence Y × x: Y ⇝ X is 2 and the multiplicity of x × Y : X ⇝ Y is zero, we can modify α by an appropriate multiple of x × Y and therefore assume that mult(α) is odd and mult(α^{t}) = 0. Hence the degree of the pull-back of α^{t} on X_{F (Y)} is zero. By [5, prop. 2.6] or [10], 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 Y \ 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 [2, 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, mult(α′) = mult(α), hence mult(α′) is odd. Since α′ is an integral linear combination of prime correspondences, we can find a prime correspondence β: X ⇝ Y′ of odd multiplicity, 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(Y) = dim(X) = dim_{es}(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 mult(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 1.2, 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 multiplicity 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 multiplicity correspondence β: Y ⇝ X′. We compose β with the embedding X′ ↪ X to produce an odd multiplicity (in fact, of the same multiplicity 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 multiplicity 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 multiplicity prime correspondences α: X ⇝ Y and γ: Y ⇝ X, we are going to construct an odd multiplicity correspondence δ: X ⇝ X with even mult(δ^{t}) getting this way a contradiction with

Theorem 3.2 ([6, th. 6.4]). Let X be an anisotropic quadric with i(X) = 1. Then for every correspondence δ : X ⇝ X, one has mult(δ) ≡ mult(δ^{t}) (mod 2).

Note that in the case where Y is smooth we can simply take for δ the composite of the correspondences α and γ (cf. [6, 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 S be a projective variety over F which is a 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 multiplicity

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

is odd and the multiplicity 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 1.2. By Lemma 2.3, the pull-back β: X′ ⇝ Y of α with respect to the embedding of X′ × Y into X × Y is an odd multiplicity 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). □

As the first corollary of the main theorem we get 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:

Corollary 3.4. Let X be an anisotropic projective F -quadric and let Y be a complete F -variety with all closed points of even degree. If dim_{es}(X) > dim(Y), then there are no rational morphisms X → Y. □

Remark 3.5. 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 mult(α^{t}) is odd for every odd multiplicity correspondence α: X ⇝ Y.

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

Corollary 3.6. 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 multiplicity correspondence X ⇝ Y which in turn gives an odd multiplicity correspondence X′ ⇝ Y. We may assume that the latter correspondence is prime and take 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 multiplicity correspondence X′ ⇝ Y′, therefore dim(Y′) ≥ dim(X′) by Theorem 3.1, and the condition (i) of Corollary 3.6 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 3.6 is satisfied. Finally the quadric X′_{F (Y′)} is isotropic by Theorem 3.1; therefore X_{F (Y′)} is isotropic. □

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 purely transcendental extension of the base field by Proposition 1.2). 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. □

We have the following upper bound for the Witt index of Y over F (X).

Corollary 4.2. Let X and Y be anisotropic quadrics over F and suppose that Y is isotropic over F (X). Then

i_{W}(Y_{F (X)}) − i(Y) ≤ dim_{es}(Y) − dim_{es}(X) .

Proof. If dim_{es}(X) = 0, the statement is trivial. Otherwise, let Y′ be a subquadric of Y of dimension dim_{es}(X) − 1. Since dim_{es}(Y′) ≤ dim(Y′) < dim_{es}(X), the quadric Y′ remains anisotropic over F (X) by the first part of Theorem 4.1. Therefore, by Lemma 1.1(3), i_{W}(Y_{F (X)}) ≤ codim_{Y} (Y′) = dim(Y) − dim_{es}(X) + 1, whence the inequality. □

Let φ be an anisotropic quadratic form over F of dimension at least 2 and let X be the quadric given by φ. A field extension K/F is called a generic splitting field of X (and of φ) if X is isotropic over K and for every field extension L/F with X_{L} isotropic, there is an F -place from K to L. In [7, §4] M. Knebusch raised the problem to determine the smallest transcendence degree of a generic splitting field of a given quadric X (we thank J. Arason for pointing out this question).

Theorem 4.3. The smallest transcendence degree of a generic splitting field of a quadric X is equal to dim_{es}(X).

Proof. Let X′ be a subquadric of X of dimension dim_{es}(X). Since X′ has a point over every field extension L/F with X_{L} isotropic, the field F (X′) is a generic splitting field of X of transcendence degree dim_{es}(X).

Let K/F be a generic splitting field of the quadric X. We will show that

(2)  tr. deg(K/F) ≥ dim_{es}(X).

Replace K by a finitely generated subextension of K/F that splits X. Let Y′ be a projective variety over F which is a model of the field extension K/F. Since X is isotropic over K, there is a rational morphism Y′ → X. Let Y be the closure of the graph of this morphism. Then Y is also a projective model of K/F and every closed point of Y has even degree since there is a morphism Y → X. Since X is isotropic over F (X), there is a place from K to F (X) over F. The valuation ring of this place has a center y ∈ Y and therefore, the place gives a morphism Spec F (X) → Y with the image {y}. In particular, Y has a rational point over F (X). By Theorem 3.1, dim(Y) ≥ dim_{es}(X), whence the inequality (2). □

We write W (F) for the Witt ring of a field F and I for the fundamental ideal of W (F), i.e., for the ideal of classes of even dimensional forms. For every n ≥ 1 the ideal I^{n} is additively generated by the classes of n-fold Pfister forms. If φ is a nonzero anisotropic form such that φ ∈ I^{n} then by the Arason-Pfister theorem [9, th. 5.6], dim(φ) ≥ 2^{n}. The following theorem was proved by A. Vishik. We give a simpler proof of this statement due to D. Hoffmann.

Theorem 4.4. Let φ be an anisotropic quadratic form such that φ ∈ I^{n} and dim(φ) > 2^{n}. Then dim(φ) ≥ 2^{n} + 2^{n−1}.

Proof. We can write the class of φ in W (F) in the form φ = φ_{1} +φ_{2} +⋯+φ_{m}, where each φ_{i} is similar to an anisotropic n-fold Pfister form. Assume that the statement is not true (in particular, n ≥ 2) and choose a counterexample φ of the smallest dimension with the smallest number m of the φ_{i}’s (over all fields).

Since the anisotropic part φ′ of the form φ_{F (φ)} has dimension smaller than φ, we must have dim(φ′) ≤ 2^{n}. Then for the quadric Y given by φ we have

i(Y) ≥ (dim(φ) − 2^{n})/2 = (dim(Y) − 2^{n})/2 + 1

and therefore,

(3) dim_{es}(Y) = dim(Y) − i(Y) + 1 ≤ dim(Y)/2 + 2^{n−1} < 2^{n} + 2^{n−2} − 1

since by assumption dim(φ) < 2^{n} + 2^{n−1}.

We are going to prove that all the φ_{i} are pairwise similar. Assume that there are two non-similar forms φ_{i} and φ_{j}. Consider the anisotropic part ψ of the form φ_{i}⊥φ_{j}. Note that ψ is not similar to an n-fold Pfister form by the choice of m.

We claim that dim(ψ) ≥ 2^{n} +2^{n−1}. By a theorem of Elman-Lam [1, th. 4.5], the Witt index of the form φ_{i}⊥φ_{j} is either 0 of 2^{s}, where s is the linkage number of φ_{i} and φ_{j}. We have s ≠ n since the forms φ_{i} and φ_{j} are non-similar and s ≠ n − 1 since ψ is not similar to an n-fold Pfister form. Thus, s ≤ n − 2 and therefore, dim(ψ) ≥ 2^{n} + 2^{n−1}. The claim is proved.

It follows from [1, th. 3.2] that ψ is not similar to an (n+1)-fold Pfister form. Therefore, by [9, th. 5.4], ψ is not hyperbolic over F (ψ). The Arason-Pfister theorem implies that the dimension of the anisotropic part of ψ_{F (ψ)} is at least 2^{n}. Hence i(ψ) ≤ (dim(ψ) − 2^{n})/2 and therefore, for the quadric X given by ψ we have

(4) dim_{es}(X) = dim(ψ) − i(ψ) − 1 ≥ dim(ψ)/2 + 2^{n−1} − 1 ≥ 2^{n} + 2^{n−2} − 1.

It follows from Theorem 4.1, (3), and (4) that φ is anisotropic over F (ψ), so that φ over F (ψ) is also a minimal counterexample. But the dimension of the anisotropic part of ψ (i.e., of φ_{i}⊥φ_{j}) over F (ψ) is smaller than dim(ψ).

Iterating the construction we come to the situation when the dimension of the anisotropic part of φ_{i}⊥φ_{j} becomes smaller than 2^{n} + 2^{n−1}, a contradiction to the claim.

We have proved that all the φ_{i} are similar to some n-fold Pfister form ρ, so that the class of φ in the Witt ring is divisible by ρ and therefore dim(φ) is divisible by 2^{n} [9, th. 5.4(iv)] since φ is anisotropic. This contradicts the inequality 2^{n} < dim φ < 2^{n} + 2^{n−1}. □

References

[1] R. Elman and T. Y. Lam, Pfister forms and K-theory of fields, J. Algebra 23 (1972), 181–213.

[2] W. Fulton, Intersection Theory, Springer-Verlag, Berlin, 1984.

[3] O. T. Izhboldin, Fields of u-invariant 9, Ann. Math. 154 (2001), no. 3, 529–587.

[4] O. T. Izhboldin, Virtual Pfister neighbors and first Witt index, Preprint (2000).

[5] N. A. Karpenko, Algebro-geometric invariants of quadratic forms, Algebra i Analiz 2 (1990), no. 1, 141–162 (in Russian); English translation: St. Petersburg Math. J. 2 (1991), no. 1, 119–138.

[6] N. A. Karpenko, On anisotropy of orthogonal involutions, J. Ramanujan Math. Soc. 15 (2000), no. 1, 1–22.

[7] M. Knebusch, Generic splitting of quadratic forms. I, Proc. London Math. Soc. (3), 33 (1976), no. 1, 65–93.

[8] T. Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin, Reading, Massachusetts, 1973.

[9] W. Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

[10] R. G. Swan, Zero cycles on quadric hypersurfaces, Proc. Amer. Math. Soc. 107 (1989), no. 1, 43–46.

[11] A. Vishik, Direct summands in the motives of quadrics, Preprint, 1999.

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