TORSION ORDERS OF FANO HYPERSURFACES STEFAN SCHREIEDER Abstract.

TORSION ORDERS OF FANO HYPERSURFACES

STEFAN SCHREIEDER

Abstract. We find new lower bounds on the torsion orders of very general Fano hy- persurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have very large degree.

Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending a previous result of the author from characteristic different from two to arbitrary characteristic.

1. Introduction

Let X be a projective variety over a field k. The torsion order Tor(X) of X is the smallest positive integer e, such that e times the diagonal of X admits a decomposition in the Chow group of X × X, that is,

e∆

= [z × X] + B ∈ CH

(X × X),

where z ∈ CH

(X) is a zero-cycle of degree e and B is a cycle on X × X that does not dominate the second factor. If no such decomposition exists, we put Tor(X) = ∞. If X is smooth and k is algebraically closed, then Tor(X) is the smallest positive integer e such that for any field extension L of k, the kernel of the degree map CH

(X

) → Z is e-torsion, and Tor(X) = ∞ if no such integer exists. This notion goes back to Bloch [Blo79] (using an idea of Colliot-Th´ el` ene) and Bloch–Srinivas [BS83], and has for instance been studied in [ACTP13] and [Voi15], and in the above form by Chatzistamatiou–Levine [CL17] and Kahn [Kah17].

The torsion order is a stable birational invariant of smooth projective varieties; it is finite if X is rationally connected and it is 1 if X is stably rational. Moreover, if f : Y → X is a generically finite morphism, then Tor(X) divides deg(f) · Tor(Y ). In particular, the degree of any unirational parametrization of X is divisible by Tor(X).

The torsion order is a powerful invariant of rationally connected varieties, which we would like to compute for interesting classes of varieties. In particular, it is desirable to

Date: June 9, 2020.

2010 Mathematics Subject Classification. primary 14J70, 14C25; secondary 14M20, 14E08.

Key words and phrases. Hypersurfaces, algebraic cycles, unirationality, rationality, unramified cohomology.

1

do so for smooth hypersurfaces X

⊂ P

of degree d ≤ N + 1. By a result of Roitman [Roi72] and Chatzistamatiou–Levine [CL17, Proposition 5.2], we have

Tor(X

) | d! . (1)

This yields an upper bound which holds over any field k.

Finding lower bounds for Tor(X

) over algebraically closed fields is in general a difficult problem. By a result of Chatzistamatiou–Levine [CL17, Theorem 8.2], building on earlier work of Totaro [Tot16] and Koll´ ar [Kol95], the torsion order of a very general complex hypersurface X

⊂ P

of degree d ≥ p

d

e is divisible by p

, where p denotes a prime number, and where p is odd or N is even. This yields non-trivial lower bounds roughly in degrees d >

N . In [Sch19b], the author dealt with lower degrees by showing that the torsion order of a very general hypersurface X

⊂ P

of degree d ≥ log

N + 2 and dimension N ≥ 3 is divisible by 2. This paper generalizes that result as follows.

Theorem 1.1. Let k be an uncountable field. Then the torsion order of a very general Fano hypersurface X

⊂ P

of degree d ≥ 4 is divisible by every integer m ≤ d− log

N that is invertible in k.

The first new case concerns very general quintic fourfolds X

⊂ P

over (algebraically closed) fields of characteristic different from 3, for which we get 3 | Tor(X

). If char k = 0, then Tor(X

) is also divisible by 2 and 5 (see [Tot16] and [CL17, Theorem 8.2]) and so our result determines all prime factors of Tor(X

) by (1).

The strength of Theorem 1.1 lies in its asymptotic behaviour for large N . To illustrate this, let X

⊂ P

be a very general complex hypersurface of degree 100. Then Tor(X

) is divisible by

2

· 3

· 5

· 7 · Y

p≤89 p prime

p = 718 766 754 945 489 455 304 472 257 065 075 294 400,

while it was previously only known to be divisible by 2

· 3

· 5

· 7 · 11 = 138 600.

Even though smooth hypersurfaces X

⊂ P

of degree d with d! ≤ log

(N + 1) are known to be unirational [HMP98, BR19], very general Fano hypersurfaces of large degree are conjecturally not unirational. While this paper does not solve this problem, it does show that for most Fano hypersurfaces, unirational parametrizations need to have enor- mously large degree, strengthening previous bounds on this problem: In [Kol95, Theorem 4.3], Koll´ ar gave lower bounds on the degree of a uniruled parametrization of high-degree Fano hypersurfaces, and, relying on [Tot16], Chatzistamatiou–Levine produced slightly better bounds for unirational parametrizations in [CL17, Theorem 8.2].

In [Sch19b] it was shown that very general hypersurfaces of dimension N ≥ 3 and

degree d ≥ log

N + 2 are stably irrational over any uncountable field of characteristic

different from two. This improved [Kol95, Tot16] in characteristic 6= 2, but the Koll´ ar–

Totaro bound d ≥ 2d

e remained the best known result in characteristic two.

Applying Theorem 1.1 to m = 3, this paper solves the rationality problem for hyper- surfaces in characteristic two under a logarithmic degree bound.

Corollary 1.2. Let k be an uncountable field of characteristic two. Then a very general hypersurface X ⊂ P

of degree d ≥ log

N + 3 is stably irrational.

The method of this paper is flexible and applies to other types of varieties as well. To illustrate this, we include here the example of cyclic covers of projective space.

Theorem 1.3. Let k be an uncountable field and let m ≥ 2 be an integer that is invertible in k. Then the torsion order of a cyclic m : 1 cover X → P

branched along a very general hypersurface of degree d ≥ m(d

e + 2) (with m | d) is divisible by m.

In particular, under the above degree bound, very general cyclic m : 1 covers are stably irrational. Even for k = C , this extends previous results on this problem substantially, see [Kol96, CTP16b, Oka19]. For m = 2, a similar result has previously been proven in [Sch19b, Theorem 9.1].

The above results are proven via a version of the degeneration technique that the author developed in [Sch19a, Sch19b] and which improved the method of Voisin [Voi15]

and Colliot-Th´ el` ene–Pirutka [CTP16a]. An essential ingredient in this approach is the construction of varieties that have nontrivial unramified cohomology.

Constructing rationally connected varieties with nontrivial unramified cohomology is a subtle problem. In degree two, the first examples are due to Saltman [Sal84]. Build- ing on [AM72], the first examples in degree three and with Z /2-coefficients have been constructed in [CTO89]. This has later been generalized to arbitrary degrees and Z /`- coefficients for any prime ` in [Pey93, Aso13]. Starting with [CTO89], all these con- structions rely on norm varieties attached to symbols in Milnor K-theory mod `. Norm varieties attached to symbols of length two are Brauer–Severi varieties. For ` 6= 2, such varieties have large degree, compared to their dimensions, which hints that they are not useful for our purposes. Moreover, for symbols of length at least three, norm varieties for ` 6= 2 are very intricate objects, whose construction, due to Rost, relies inductively on the Bloch–Kato conjecture in lower degrees, see [SJ06]. The situation is special for

` = 2, where norm varieties are Pfister quadrics, which are much simpler objects. Pfister quadrics are used in [Sch19b], which explains the restriction to the prime 2.

This paper introduces for any integer m large classes of hypersurfaces with unramified

Z /m-cohomology, see Theorem 5.3 below. As in [Sch19b], an important ingredient is

a quite flexible degeneration argument which allows to prove nontriviality of certain

classes without any deep result from K-theory, see item (3) in Theorem 5.3. Besides the

ideas from [Sch19b], the main new ingredient of this paper is the definition and usage of universal relations in Milnor K-theory, see Definition 3.1 below. Our approach is elementary, works for any positive integer m (not necessarily prime) and does not rely on norm varieties, nor on Voevodsky’s proof of the Bloch–Kato conjectures.

Asok’s examples [Aso13] with nontrivial unramified Z /`-cohomology in degree n have dimension N `

, which grows rapidly with n. For any given prime ` and integer N ≥ 3, this led Asok [Aso13, Question 4.5] to ask for general restrictions on the pos- sible degrees in which rationally connected complex varieties of dimension N can have nontrivial unramified Z /`-cohomology. This is a quite subtle problem already for ratio- nally connected threefolds, where by a result of Voisin [Voi06] and Colliot-Th´ el` ene–Voisin [CTV12, Th´ eor` eme 1.2], it boils down to understanding the possible Brauer groups, see [Aso13, Remarks 4.7, 4.8 and 4.10].

As a consequence of our proof of Theorem 1.1, we obtain the following uniform result in arbitrary dimension; the case m = 2 is due to [Sch19b, Theorem 1.5].

Theorem 1.4. Let m, n ≥ 2 and N ≥ 3 be integers with log

(m + 1) ≤ n ≤ N + 1 − m.

Then there is a rationally connected smooth complex projective variety X of dimension N such that the n-th unramified cohomology H

( C (X)/ C , Z /m) of X contains an element of order m.

It is natural to wonder whether the upper bound in the above theorem is sharp. For instance, is the unramified Z /2-cohomology of a rationally connected smooth complex projective variety X trivial in degree n = dim X?

Remark 1.5. The main results of this paper are formulated over uncountable fields.

However, our proofs show that one can also write down examples over small fields, see e.g. Section 9 below for some explicit examples over Q (t) and F

(t, s). Moreover, the varieties in Theorem 1.4 may be chosen to be defined over Q , see Remark 7.3 below.

2. Preliminaries

2.1. Conventions. A variety is an integral separated scheme of finite type over a field.

For a scheme X, we denote its codimension one points by X

. A property holds for a very general point of a scheme if it holds at all closed points inside some countable intersection of open dense subsets.

2.2. Degenerations. Let R be a discrete valuation ring with fraction field K and alge-

braically closed residue field k. Let X → Spec R be a proper flat morphism with generic

fibre X and special fibre Y . Then we say that X degenerates to Y . We also say that

the base change of X to any larger field degenerates (or specializes) to Y . For instance,

if X → B is a proper flat morphism of varieties over an algebraically closed uncountable

field, then the fibre X

over a very general point t ∈ B degenerates to the fibre X

for any closed point 0 ∈ B in the above sense, see e.g. [Sch19a, §2.2]. In particular, a very general hypersurface X ⊂ P

over an algebraically closed uncountable field k specializes to any given hypersurface of the same dimension and degree over k.

2.3. Alterations. Let Y be a variety over an algebraically closed field k. An alteration of Y is a proper generically finite surjective morphism τ : Y

→ Y , where Y

is a non- singular variety over k. De Jong [deJ96] proved that alterations always exist. Later, Gabber showed that one can additionally require that deg(τ) is prime to any given prime ` 6= char(k). Temkin [Tem17, Theorem 1.2.5] generalized this further, ensuring that deg(τ ) is a power of the characteristic of k (or one if char(k) = 0).

2.4. Milnor K-theory. Let L be a field. Recall that Milnor K-theory K

(L) of L in degree n ≥ 2 is defined as the quotient of (L

)

, where L

denotes the multiplicative group of units in L, by the subgroup generated by tensors of the form a

⊗ · · · ⊗ a

with a

+ a

= 1 for some 1 ≤ i ≤ n − 1. Moreover, K

(L) = Z and K

(L) = L

. The image of a tensor a

⊗ · · · ⊗ a

in K

(L) is denoted by (a

, . . . , a

). The direct sum K

(L) := L

n≥0

K

(L) has a natural product structure, induced by the tensor product.

For an integer m ≥ 2, the defining relation for Milnor K-theory implies the following basic relation in Milnor K-theory mod m, see [Mil70, Lemma 1.3].

Lemma 2.1. Let L be a field and let b

, . . . , b

∈ L

such that P

b

= c

for some c ∈ L.

Then (b

, . . . , b

) = 0 ∈ K

(L)/m.

Let A be a ring and let A

be the multiplicative group of units in A. We define K

(A) as the quotient of (A

)

by the subgroup generated by a

⊗ · · · ⊗ a

with a

+ a

= 1 for some 1 ≤ i ≤ n − 1. If A is a field, then this definition coincides with the one above.

If A → B is a homomorphism of rings, then we obtain an induced homomorphism K

(A) → K

(B ). In particular, if A is an integral domain with fraction field L, then there is a natural map ψ

: K

(A) → K

(L), and if A is local with residue field κ, then there is a natural map ψ

: K

(A) → K

(κ). The following result is known as specialization property in Milnor K-theory, c.f. [CT95, Definition 2.1.4.c].

Lemma 2.2. Let A be a regular local ring with fraction field L and residue field κ. Then for any integer m ≥ 1, we have

ker ψ

: K

(A)/m ^// K

(L)/m

⊆ ker ψ

: K

(A)/m ^// K

(κ)/m .

Proof. By [CT95, Lemma 2.1.5(a)], it suffices to prove the lemma in the case where A is

a complete discrete valuation ring. In this case, let π ∈ A be a uniformizer. This induces

a residue homomorphism ∂

: K

(L)/m → K

(κ)/m (which in fact depends only on A and not on the choice of the uniformizer π), such that for α ∈ K

(A)/m,

ψ

(α) = ∂

((π) ⊗ ψ

(α)),

where (π) ∈ K

(L)/m = L

/(L

)

, see [Mil70, Lemma 2.1]. This immediately shows

ker(ψ

) ⊂ ker(ψ

), as we want.

2.5. Galois cohomology and unramified cohomology. Let m ≥ 2 be a positive integer and let L be a field in which m is invertible. We denote by µ

⊂ L the ´ etale sheaf of m-th roots of unity. For an integer j ≥ 1, we consider the twists µ

:= µ

⊗ · · · ⊗ µ

(j-times) and define µ

:= Z /m and µ

:= Hom(µ

, Z /m) for j < 0. For a field L in which m is invertible and which contains a primitive m-th root of unity, there is, for any integer j, an isomorphism µ

(Spec L) ' Z /m.

Let L be a field and let m be an integer that is invertible in L. For any integer j, we denote by H

(L, µ

) the Galois cohomology of the absolute Galois group of L with the natural action on µ

(Spec L

), where L

denotes a separable closure of L.

Kummer theory induces a canonical isomorphism H

(L, µ

) ' L

/(L

)

= K

(L)/m.

By [Tat76, Theorem 3.1], this induces via cup products a morphism of graded rings K

(L)/m ^// H

(L, µ

) := M

i≥0

H

(L, µ

).

(2)

(In fact, this map is an isomorphism by the Bloch–Kato conjecture, proven by Voevodsky, but we will not use this fact in this paper.) By slight abuse of notation, we denote the image of a class (a

, . . . , a

) ∈ K

(L)/m in H

(L, µ

) by the same symbol.

Let A be a ring with

∈ A and let L be a field. Let A → L be a homomorphism of rings. Since H

(L, µ

) coincides with the ´ etale cohomology of Spec L with values in the

´ etale sheaf µ

, there is a natural pullback map H

(Spec A, µ

) → H

(L, µ

). This applies in particular to the case where A with

∈ A is a local ring with residue field L, or an integral domain with fraction field L. By [CT95, Lemma 2.1.5(b) and §3.6], the following specialization property (c.f. [CT95, Definition 2.1.4.c]), analogues to Lemma 2.2, holds in ´ etale cohomology.

Lemma 2.3. Let A be a regular local ring with fraction field L and residue field κ. Then for any integer m ≥ 1 that is invertible in κ, we have

ker H

(Spec A, µ

) → H

(L, µ

)

⊆ ker H

(Spec A, µ

) → H

(κ, µ

) . For any discrete valuation ν on a field L, such that m is invertible in the residue field κ(ν), there is a residue map

∂

: H

(L, µ

) ^// H

(κ(ν), µ

),

which (for i = j) is compatible with the aforementioned residue map in Milnor K-theory.

This map is surjective and its kernel is described as follows, see e.g. [CT95, (3.10)].

Theorem 2.4. Let ν be a discrete valuation on a field L such that m is invertible in the residue field κ(ν). Let O

⊂ L be the associated discrete valuation ring. Then the natural sequence

0 ^// H

(Spec O

, µ

) ^// H

(L, µ

)

^// H

(κ(ν), µ

) ^// 0.

(3) is exact.

Assume now that L = k(X) is the function field of a k-variety X and let Val(L/k) be the set of all valuations on L that are induced by a prime divisor on some normal birational model of X. The unramified µ

-cohomology of X in degree i is defined as

H

(k(X)/k, µ

) := {α ∈ H

(k(X), µ

) | ∂

α = 0 ∀ν ∈ Val(k(X)/k)}.

This subgroup of H

(k(X), µ

) is a stable birational invariant of X, see [CTO89]. If k contains a primitive m-th root of unity, then µ

' µ

= Z /m for all j and so H

(k(X)/k, µ

) ' H

(k(X)/k, Z /m).

Let γ ∈ H

(k(X)/k, µ

) be unramified and let E ⊂ X be a subvariety whose generic point x lies in the smooth locus of X. Then γ lifts uniquely to a class in the cohomology of Spec O

(see e.g. [CT95, Theorem 3.8.2]) and so it can be restricted to the closed point to give a class in H

(κ(x), µ

) = H

(k(E), µ

) that we denote by γ|

or γ|

.

3. Universal relations in Milnor K-theory modulo m Fix a base field k and a natural number m ≥ 2. For integers n, s ≥ 1, let

R

:= k[x

, x

, . . . , x

, y

, . . . , y

]

be the polynomial ring over k in n + s variables and let L

:= Frac R

be its field of fractions.

Definition 3.1. A universal relation in Milnor K-theory modulo m over the field k is an identity

(x

, . . . , x

) = λ · (a

, . . . , a

) ∈ K

(L

)/m, (4)

for some nonzero polynomials a

, . . . , a

∈ R

and λ ∈ ( Z /m)

.

3.1. General properties.

Lemma 3.2. Let (4) be a universal relation in Milnor K-theory modulo m over the field k. Let L be a field extension of k and let φ : R

→ L be a morphism of k-algebras such that φ(x

) and φ(a

) are invertible in L for all i = 1, . . . , n. Then,

(φ(x

), . . . , φ(x

)) = λ · (φ(a

), . . . , φ(a

)) ∈ K

(L)/m.

Proof. The morphism φ yields a morphism of schemes ϕ : Spec L → Spec R

= A

. Let x ∈ A

be the image of ϕ. Then the field L is an extension of the residue field κ(x) of x and so there is a natural homomorphism K

(κ(x))/m → K

(L)/m. In order to prove the lemma, we may thus without loss of generality assume L = κ(x) and so ϕ denotes the inclusion of the scheme-point x ∈ A

.

Let A be the local ring of A

at x. Since φ(x

), φ(a

) ∈ L

, we get x

, a

∈ A

and so (x

, . . . , x

) − λ(a

, . . . , a

) ∈ K

(A)/m.

(5)

This element lies in the kernel of K

(A)/m → K

(Frac A)/m, because Frac A = L

and (4) is a universal relation. It thus follows from Lemma 2.2 that (5) lies in the kernel of K

(A)/m → K

(L)/m, because L = κ(x) is the residue field of the local ring A by

assumption. This concludes the lemma.

Remark 3.3. The above lemma explains the terminology in Definition 3.1. Indeed, let (4) be a universal relation, let L be any field extension of k and let (χ

, . . . , χ

) ∈ K

(L)/m be any symbol. By the universal property of polynomial rings, we may then define φ : R

→ L by setting φ(x

) = χ

for i = 1, . . . , n and φ(y

) ∈ L arbitrary such that φ(a

) 6= 0 for all i = 1, . . . , n. By Lemma 3.2, we get a relation

(χ

, . . . , χ

) = λ · (φ(a

), . . . , φ(a

)) ∈ K

(L)/m in the Milnor K-theory of L, which involves the symbol (χ

, . . . , χ

).

The following proposition shows that universal relations allow us to construct varieties whose function fields kill a given symbol in Milnor K-theory modulo m.

Proposition 3.4. Let (4) be a universal relation in Milnor K-theory modulo m over the field k in degree n ≥ 1. Let L be a field extension of k and let χ

, . . . , χ

∈ L

. Let s

be a positive integer and let φ : R

→ L[y

, . . . , y

] a homomorphism of k-algebras with φ(x

) = χ

for all i = 1, . . . , n. Let c ∈ L[y

, . . . , y

] be such that

F := c

−

n

X

i=1

φ(a

) ∈ L[y

, . . . , y

] is irreducible and let W be a projective model of {F = 0} ⊂ A

0

L

. Assume that for each

i, φ(a

) is not a multiple of F , i.e. the restriction of φ(a

) to W is nonzero. Then

(a) (χ

, . . . , χ

) ∈ ker K

(L)/m ^// K

(L(W ))/m .

(b) Let Y be a variety over L and let ι : Y → W be a morphism of L-varieties such that the image ι(η

) of the generic point of Y lies in the regular locus of W . Then

(χ

, . . . , χ

) ∈ ker K

(L)/m ^// K

(L(Y ))/m .

Proof. Since F is irreducible, W is integral and so it is regular at the generic point. In particular, item (a) is a special case of (b). Nonetheless, we will prove (a) first. For this, we denote by φ(a

) the image of a

∈ R

in L(W ). By our assumptions, φ(a

) 6= 0 for all i. Hence,

(φ(x

), . . . , φ(x

)) = λ · (φ(a

), . . . , φ(a

)) ∈ K

(L(W ))/m by Lemma 3.2, and so this class vanishes by Lemma 2.1 because P

i

φ(a

) is an m-th power in L(W ) by the definition of F . This proves item (a) because φ(x

) = χ

for all i.

To prove item (b), let w = ι(η

) ∈ W be the image of the generic point of Y . Let A = O

be the local ring of W at w. By assumption, A is a regular local ring. Since χ

∈ L

⊂ A

for all i,

(χ

, . . . , χ

) ∈ ker(K

(A)/m ^// K

(L(W ))/m) by item (a) proven above. Applying Lemma 2.2, we then find

(χ

, . . . , χ

) ∈ ker(K

(A)/m ^// K

(κ(w))/m).

Item (b) stated in the proposition follows from this because L(Y ) is a field extension of κ(w) and so the natural map K

(L)/m ^// K

(L(Y ))/m factors through K

(κ(w))/m.

This concludes the proof of the proposition.

3.2. Examples. The simplest example of a universal relation modulo m is given by (x

) = (x

y

) ∈ K

(L

)/m.

The next lemma allows to produce universal relations in Milnor K-theory mod m in arbitrary degree by starting with a single relation in low degree.

Lemma 3.5. Let (x

, . . . , x

) = λ · (a

, . . . , a

) ∈ K

(L

)/m be a universal relation in degree n. Then

(x

, . . . , x

, x

) = λ · a

, . . . , a

, x

y

−

n

X

i=1

a

!!

∈ K

(L

)/m,

is a universal relation in degree n + 1, where a

:= a

(x

, . . . , x

, y

, . . . , y

).

Proof. Since (x

, . . . , x

) = λ · (a

, . . . , a

) ∈ K

(L

)/m, we have x

, . . . , x

, x

y

−

n

X

i=1

a

!!

= λ · a

, . . . , a

, x

y

−

n

X

i=1

a

!!

in K

(L

)/m. The claim in the lemma is thus equivalent to x

, . . . , x

, y

−

n

X

i=1

a

!

= 0 ∈ K

(L

)/m.

(6)

Relabelling the y-coordinates in the universal relation (x

, . . . , x

) = λ · (a

, . . . , a

) shows by Lemma 3.2 that (x

, . . . , x

) = λ · (a

, . . . , a

) ∈ K

(L

)/m and so (6) is equivalent to

λ · a

, . . . , a

, y

−

n

X

i=1

a

!

= 0 ∈ K

(L

)/m,

which holds by Lemma 2.1. This concludes the proof of the lemma.

To illustrate the above result, start with the trivial relation (x

) = (x

y

) in degree one. Applying the lemma, we arrive at the relation

(x

, x

) = (x

y

, x

y

− x

x

y

) ∈ K

(L

)/m

in degree two. Applying the lemma once again, we get the universal relation (x

, x

, x

) = (x

y

, x

y

− x

x

y

, x

y

− x

x

y

− x

x

y

+ x

x

x

y

) in K

(L

)/m. Repeating this process inductively, we are led to the universal relation in degree n from Proposition 4.1 below.

4. Fermat–Pfister forms

Let k be a field and m ≥ 2 an integer. For n ≥ 1, we define the n-th Fermat–Pfister form of degree m (and with coefficients in the polynomial ring k[x

, . . . , x

]) as

Pf

(y

, . . . , y

) := X

∈{0,1}ⁿ

(−x

)

(−x

)

. . . (−x

)

· y

, (7)

where φ : {0, 1}

→ {0, 1, . . . , 2

− 1} denotes the bijection given by φ() =

n

X

i=1

· 2

.

This generalizes the famous quadratic forms of Pfister [Pfi65] to higher degrees.

We denote the coefficient in front of y

by c

and get

Pf

(y

, . . . , y

) =

2ⁿ−1

X

j=0

c

y

. (8)

By definition, c

= 1, c

= −x

and c

= (−1)

x

· · · x

. For n ≥ 1, we have

Pf

(y

, . . . , y

) = Pf

(y

, . . . , y

) − x

· Pf

(y

, . . . , y

), where we set Pf

(y

) := y

. Inductively, this yields

Pf

(y

, . . . , y

) = y

−

n

X

i=1

a

, (9)

where

a

:= x

· Pf

(y

, . . . , y

).

(10)

Proposition 4.1. Let k be a field and let a

∈ k[x

, . . . , x

, y

, . . . , y

] be as in (10).

Then,

(x

, . . . , x

) = (a

, . . . , a

) ∈ K

(L

)/m, is a universal relation in Milnor K-theory modulo m over k.

Proof. We aim to prove the proposition by induction on n. For n = 1, the proposition is saying that (x

) = (x

y

), which is clear. We now assume that the proposition is proven for some n ≥ 1 and we aim to prove it for n + 1. Applying Lemma 3.5 to the given universal relation in degree n, we obtain

(x

, . . . , x

, x

) = a

, . . . , a

, x

y

−

n

X

i=1

a

!!

, (11)

in K

(L

)/m, where

a

= a

(x

, . . . , x

, y

, . . . , y

) = x

· Pf

(y

, . . . , y

).

The recursive relation (9) implies y

−

n

X

i=1

a

= Pf

(y

, y

, . . . , y

) and so

x

y

−

n

X

i=1

a

!

= x

Pf

(y

, y

, . . . , y

) = a

1

After this paper was submitted, it was brought to my attention that such generalizations of Pfister

forms may already be found in a paper by Krashen and Matzri [KM15, Proposition 1.9].

by (10). Hence, (11) simplifies to

(x

, . . . , x

, x

) = (a

, . . . , a

, a

) ∈ K

(L

)/m,

as we want. This concludes the proposition.

As an interesting example, the above discussion allows us to prove the following result, c.f. the aforementioned paper [KM15, Proposition 1.9] by Krashen and Matzri for a similar result, proven with different methods.

Corollary 4.2. Let n ≥ 2 be an integer and let χ

, . . . , χ

∈ L be nonzero elements of a field L. Consider the hypersurface X

⊂ P

n−1

L

of degree m, given by X

∈{0,1}ⁿ

(−χ

)

(−χ

)

. . . (−χ

)

· y

= 0, where φ() = P

i=1

· 2

. If X

is integral (e.g. if

∈ L), then (χ

, . . . , χ

) ∈ ker(K

(L)/m ^// K

(L(X

))/m).

Proof. Let k be the prime field of L and consider the polynomial ring R

= k[x

, . . . , x

, y

, . . . , y

]

from Section 3. Let φ : R

→ L[y

, y

, . . . , y

] be the morphism of k-algebras, given by φ(x

) = χ

and φ(y

) = y

for all i and j . Let further a

∈ R

be as in (10), so that the universal relation (x

, . . . , x

) = (a

, . . . , a

) ∈ K

(L

)/m holds by Proposition 4.1. By (9), the hypersurface X

from Corollary 4.2 is given by F = 0 where

F := y

−

n

X

i=1

φ(a

).

By assumption, X

is integral and so F is irreducible. Since φ(x

) = χ

∈ L

and φ(a

) 6≡ 0 mod F , it thus follows from item (a) in Proposition 3.4 that

(χ

, . . . , χ

) ∈ ker(K

(L)/m ^// K

(L(X

))/m).

This proves Corollary 4.2.

Note that in Corollary 4.2, the integer m is not assumed to be invertible in L. Adding this assumption, X

is automatically integral and in fact smooth over L and we obtain the following stronger statement.

Corollary 4.3. Let n ≥ 2 be an integer and let L be a field in which m is invertible and let χ

, . . . , χ

∈ L

. Consider the smooth hypersurface X

⊂ P

n−1

L

of degree m, given by

X

∈{0,1}ⁿ

(−χ

)

(−χ

)

. . . (−χ

)

· y

= 0,

where φ() = P

i=1

· 2

. Let Y be a variety over L which admits a morphism ι : Y → X

of L-varieties. Then

(χ

, . . . , χ

) ∈ ker K

(L)/m ^// K

(L(Y ))/m .

Proof. Let k be the prime field of L and recall R

= k[x

, . . . , x

, y

, . . . , y

] from section 3. Let φ : R

→ L[y

, . . . , y

] be the morphism of k-algebras with φ(x

) = χ

and φ(y

) = y

for all i = 1, . . . , n and j = 1, . . . , 2

− 1.

Note that W := X

is defined by the Fermat–Pfister form Pf

(y

, . . . , y

) of degree m from (7), where x

is replaced by χ

for i = 1, . . . , n. Hence, by (9),

W = (

y

−

n

X

i=1

φ(a

) = 0 )

⊂ P

n−1 L

,

where a

= x

Pf

(y

, . . . , y

). Recall also that (x

, . . . , x

) = (a

, . . . , a

) ∈ K

(L

)/m is a universal relation in Milnor K-theory modulo m by Proposition 4.1.

Since χ

6= 0 for all i and m is invertible in L, W = X

is smooth over L by the Jacobi criterion. In particular, the image ι(η

) ∈ W of the generic point of Y lies in the regular locus of W and so Corollary 4.3 follows from item (b) in Proposition 3.4.

5. Unramified cohomology via universal relations

Definition 5.1. Let k be a field. A homogeneous polynomial g ∈ k[x

, x

, . . . , x

] is of twisting type modulo m if for all i = 0, 1, . . . , n:

• g contains the monomials x

nontrivially;

• g is an m-th power modulo x

.

An inhomogeneous polynomial b ∈ k[x

, . . . , x

] is of twisting type modulo m if its homogenization in k[x

, x

, . . . , x

] has this property.

Note that the degree of a polynomial which is of twisting type modulo m must be a multiple of m. The following slightly technical lemma will be crucial.

Lemma 5.2. Let b ∈ k[x

, . . . , x

] be an inhomogeneous polynomial of twisting type modulo m. Let x ∈ S

be a codimension one point of some normal birational model S of P

. Let z ∈ P

be the image of x under the birational map S 99K P

and assume that z is the generic point of the intersection of c ≥ 1 coordinate hyperplanes {x

= · · · = x

= 0}

with 0 ≤ i

< · · · < i

≤ n. Then b becomes an m-th power in the fraction field of the completion O d

of the local ring of S at x.

Proof. Let g ∈ k[x

, x

, . . . , x

] be the homogeneous polynomial of twisting type given

by homogenization of b. The existence of z implies c ≤ n and so there is some index

0 ≤ i

≤ n with x

(z) 6= 0. Let b

be the inhomogeneous polynomial, given by setting x

= 1 in g. Then

b x

x

, x

x

, . . . , x

x

= g x

x

, x

x

, x

x

, . . . , x

x

= x

x

· g x

x

, x

x

, . . . , x

x

= x

x

· b

x

x

, x

x

, . . . , x c

x

, . . . , x

x

.

Since g is of twisting type modulo m, deg(g) is divisible by m and so b becomes an m-th power in Frac O d

if and only if this holds for b

. For this reason we may without loss of generality assume that i

= 0 and so x

(z) 6= 0. In particular, the inhomogenization b given by setting x

= 1 in g is a rational function on P

that is regular locally at z (e.g.

on the affine piece {x

6= 0} which contains the point z). That is, b is contained in the local ring O

⊂ k( P

) = k(x

, . . . , x

).

Since g contains the monomials x

nontrivially for all i = 0, . . . , n, we have b(0, 0, . . . , 0) 6= 0

and so b does not vanish at z, because z is the generic point of an intersection of c ≥ 1 hyperplanes. That is, the image b of b in κ(z) is nontrivial. Moreover, b is an m-th power, as g is an m-th power modulo x

for all i and c ≥ 1 by assumption. The result

thus follows from Hensel’s lemma, applied to O d

.

For n = 2 and m = 2, the equation of a conic tangent to the three coordinate lines in P

is of twisting type, see [HPT18]. An instructive example for arbitrary m and n is given by

g = G

+ x

x

· · · x

, (12)

where G is homogeneous of degree e with em > n and G contains x

nontrivially for all i = 0, 1, . . . , n. For m = 2, this simple but flexible example was used very successfully in [Sch19b]. The general idea of tangentially meeting degeneracy loci goes back to Artin–

Mumford [AM72] and has since then been used by many authors, see e.g. [CTO89, Pir18, Sch19a].

Theorem 5.3. Let m ≥ 2, n, s ≥ 1 be integers and let k be an algebraically closed

field in which m is invertible. Let (x

, . . . , x

) = λ · (a

, . . . , a

) ∈ K

(L

)/m be

a universal relation in Milnor K-theory modulo m over k and let b ∈ k[x

, . . . , x

] be

an inhomogeneous polynomial of twisting type modulo m, see Definitions 3.1 and 5.1.

Assume that the polynomial F := b −

n

X

i=1

a

∈ R

= k[x

, . . . , x

, y

, . . . , y

] (13)

is irreducible and let W be a projective model of {F = 0} ⊂ A

such that projection to the x

-coordinates induces a morphism h : W → P

. Let Y be a projective variety over k together with a morphism ι : Y → W , such that

• the image ι(η

) of the generic point of Y lies in the smooth locus of W ;

• the composition f := h ◦ ι : Y ^// P

is surjective.

Then

α := (x

, . . . , x

) ∈ H

(k( P

), µ

) has the following properties.

(1) The pullback f

α ∈ H

(k(Y )/k, µ

) is unramified over k.

(2) For any generically finite dominant morphism of k-varieties τ : Y

→ Y and any subvariety E ⊂ Y

which meets the smooth locus of Y

and which does not dominate P

via f ◦ τ , we have

(τ

f

α)|

= 0 ∈ H

(k(E), µ

).

(3) Assume that there is a discrete valuation ring R ⊂ k with residue field κ and a proper flat R-scheme Y → Spec R with Y ' Y ×

k. Assume further that f : Y → P

extends to a morphism f

: Y → P

whose base change f

: Y

:= Y ×

κ → P

to the special point of Spec R admits a rational section ξ : P

99K Y

whose image lies generically in the smooth locus of Y

. Then f

α ∈ H

(k(Y )/k, µ

) has order m, i.e., e · f

α 6= 0 for all e = 1, 2, . . . , m − 1.

Proof. Since E ⊂ Y

in item (2) meets the smooth locus of Y

, we may without loss of generality assume that Y

is normal. Replacing Y by its normalization, we may then assume that Y is normal as well (because τ : Y

→ Y factors through the normalization of Y , once Y

is normal). By the same argument as at the beginning of the proof of [Sch19b, Proposition 5.1], item (1) and (2) follow if we can show that for any codimension one point y ∈ Y

, which does not map to the generic point of P

,

∂

(f

α) = 0 ∈ H

(κ(y), µ

) and (f

α)|

= 0 ∈ H

(κ(y), µ

).

(14)

To prove (14), let us fix y ∈ Y

as above and let c denote the number of coordinate hyperplanes {x

= 0} ⊂ P

which contain the point f (y). By [Mer08, Proposition 1.6], we may also choose a normal birational model S of P

, such that y maps via the induced rational map Y 99K S to a codimension one point x ∈ S

on S.

Let us first assume that f(y) ∈ P

has codimension c. Then f(y) must be the generic

point of an intersection of c coordinate hyperplanes. (In particular, we have c ≥ 1,

because f (y) is not the generic point of P

.) Since b is of twisting type modulo m, it follows from Lemma 5.2 that b becomes an m-th power in the fraction field L := Frac O d

of the completion O d

of the local ring of S at x.

Let Y

and W

be the generic fibres of f : Y → P

and h : W → P

, respectively.

These are varieties over the field k( P

). Since L is a field extension of k( P

), we can consider the L-varieties

(Y

)

:= Y

×

L and (W

)

:= W

×

L.

Since b = (b

)

for some b

∈ L, we find that (W

)

is birational to (

(b

)

−

n

X

i=1

a

= 0 )

⊂ A

.

The morphism ι : Y → W induces a morphism (ι

)

: (Y

)

→ (W

)

and the image of the generic point of (Y

)

lies in the smooth locus of (W

)

, by assumption. Since (x

, . . . , x

) = λ · (a

, . . . , a

) ∈ K

(L

)/m is a universal relation modulo m over k, we thus deduce from Proposition 3.4, applied to the natural morphism φ : R

→ L[Y

], induced by k[x

, . . . , x

] ⊂ L, that

(x

, . . . , x

) ∈ ker K

(L)/m ^// K

(L(Y

))/m . (15)

Let now O d

be the completion of Y at the codimension one point y. Then the fraction field Frac O d

is a field extension of L(Y

) and so (15) implies

(x

, . . . , x

) ∈ ker

K

(L)/m ^// K

Frac O d

/m

.

Mapping this identity to cohomology via (2), we find that f

α lies in the kernel of the natural map

ϕ : H

(k(Y ), µ

) ^// H

(Frac O d

, µ

).

The residue of f

α at y factors through ϕ, and so ∂

f

α = 0. This implies ϕ(f

α) = 0 ∈ H

(Spec O d

, µ

) ⊂ H

(Frac O d

, µ

),

where the latter inclusion follows from Theorem 2.4. Hence, the restriction (f

α)|

factors through ϕ as well and so (f

α)|

= 0, which concludes (14) in this case.

It remains to deal with the case where f(y) ∈ P

has codimension greater than c (e.g.

this happens if c = 0). Using homogeneous coordinates, we have α =

x0

, . . . ,

0

. Fix some j ∈ {1, . . . , n}. Multiplying each entry of α by (x

/x

)

, we find

α = x

x

x

, . . . , x

x

, . . . , x

x

x

!

.

Since k is algebraically closed, (−1) ∈ (K

)

and so (a, a) = 0 for any a ∈ k( P

)

, see Lemma 2.1. Applying this to a = (x

/x

)

, the above identity simplifies to

α = x

x

, . . . , x

x

, . . . , x

x

!

= − x

x

, . . . , x

x

, . . . , x

x

.

Since it suffices to prove (14) after changing the sign of α, we may thus, up to relabelling the coordinates, without loss of generality assume that x

, . . . , x

vanish at f(y), while x

, x

, . . . , x

do not vanish at f (y).

Now the same argument as in Case 2 of the proof of [Sch19b, Proposition 5.1] applies;

we repeat it for convenience of the reader.

First recall the normal birational model S of P

, such that y maps to a codimension one point x ∈ S

on S. Since x

, x

, . . . , x

do not vanish at f(y), we get

∂

α = (∂

(x

, . . . , x

)) ∪ (x

, . . . , x

) ∈ H

(κ(x), µ

),

see e.g. [Sch19b, Lemma 2.1]. Since f(y) has codimension greater than c and k is algebraically closed, H

(κ(f (y)), µ

) = 0 for all j . Hence, (x

, . . . , x

) = 0 ∈ H

(κ(f (y)), µ

) and so ∂

α = 0 by the above formula. Since ∂

α is up to a multiple given by the pullback of ∂

α (see e.g. [CT95, Proposition 3.3.1]), we find that

∂

f

α = 0. Moreover, the restriction f

α|

is given by pulling back the restriction α|

∈ H

(κ(x), µ

), which vanishes because κ(x) has cohomological dimension less than n, since k is algebraically closed. This proves (14), which establishes items (1) and (2) of Theorem 5.3.

To prove (3), we define for any given field K, the class α

:= (x

, . . . , x

) ∈ H

(K( P

), µ

).

In particular, α = α

in the notation of Theorem 5.3. We then assume for a contradiction that for some e ∈ {1, 2, . . . , m − 1},

e · f

α

= 0 ∈ H

(k(Y ), µ

).

(16)

We denote the fraction field of R by L := Frac R and consider the morphism f

: Y → P

that extends f by assumption. In a first step, we aim to reduce to the case where

e · f

α

= 0 ∈ H

(L(Y ), µ

).

(17)

(Here, by slight abuse of notation, we use that the k-variety Y can be thought of as a variety over the smaller field L ⊂ k, because Y ' Y ×

k and L = Frac R.)

Since k is algebraically closed, H

(k(Y ), µ

) → H

(K(Y ), µ

) is injective for any

field extension K of k. (To see this, it suffices to treat the case where K is a finitely

generated field extension of k, hence the fraction field of an affine k-variety B and so the

claim follows after specialization to a k-point of B, which exists because k is algebraically

closed.) We may thus without loss of generality assume that in (16), k is the algebraic closure of Frac R. Note also that the assumptions in item (3) of Theorem 5.3 are stable under base change via an extension of discrete valuation rings R ⊂ R

. Replacing R by its completion ˆ R, Y → Spec R by the corresponding base change and k by the algebraic closure of ˆ R, we may thus assume that R is complete. Since H

(k(Y ), µ

) is the direct limit lim

H

(L

(Y ), µ

), where L

runs through all finitely generated extensions of L = Frac R, (17) shows that there is a finitely generated field extension L

of Frac R such that e · f

α maps to zero on H

(L

(Y ), µ

). Since k is the algebraic closure of Frac R, L

is in fact a finite field extension of Frac R. Replacing R by its integral closure in L

(which is again a discrete valuation ring because R is complete, see [EGAIV, Th´ eor` eme 23.1.5 and Corollaire 23.1.6]), Y → Spec R by the corresponding base change and κ by the induced finite field extension, we may finally assume that (17) holds, as we want.

By assumption, there is a rational section ξ : P

→ Y

such that the image y

= ξ(η

) of the generic point of P

is contained in the smooth locus of Y

. Since R is a discrete valuation ring and Y

is the special fibre of the proper flat morphism Y → Spec R, we find that y

is contained in a unique irreducible component Y

of Y

and Y

must be generically reduced along Y

. In particular, the local ring A := O

0

of Y at the generic point of Y

is a discrete valuation ring with fraction field L(Y ), where we recall that L = Frac R. The morphism f

: Y → P

induces a chain of inclusions L( P

) ⊂ A ⊂ L(Y ). Using this, the pullback map f

: H

(L( P

), µ

) → H

(L(Y ), µ

) factors through H

(Spec A, µ

).

It thus follows from the vanishing in (17) and the specialization property in Lemma 2.3 that the image of e · α

in H

(Spec A, µ

) restricts to zero on the special point of Spec A. That is,

e · f

α

= 0 ∈ H

(κ(Y

), µ

), (18)

where f

: Y

→ P

denotes the base change of f

: Y → P

over R to κ.

Let B be the local ring of Y

at the generic point y

= ξ(η

) of the image of the section ξ : P

99K Y

. Since the image of ξ is generically contained in the component Y

of Y

and in the smooth locus of Y

, B is a regular local ring with fraction field κ(Y

).

The morphism f

: Y

→ P

thus induces a chain of inclusions κ( P

) ⊂ B ⊂ κ(Y

).

Using this, the pullback map f

: H

(κ( P

), µ

) → H

(κ(Y

), µ

) factors through H

(Spec B, µ

). The vanishing in (18) and the specialization property in Lemma 2.3 thus imply that the image of e · α

in H

(Spec B, µ

) restricts to zero on the special point Spec κ(y

) of Spec B. However, the composition

H

(κ( P

), µ

) ^// H

(Spec B, µ

) ^// H

(κ(y

), µ

),

is an isomorphism, because y

= ξ(η

) is the generic point of the image of the ratio-

nal section ξ : P

99K Y

of f

: Y

→ P

. Hence, the aforementioned vanishing in