RATIONALITY OF BRAUER-SEVERI SURFACE BUNDLES

By

Shitan Xu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Mathematics—Doctor of Philosophy

2025

ABSTRACT

Rationality problems of complex algebraic geometry has a long history. Recent developments,

usually referred as specialization method in the literature, have given fruitful new examples of

non-rational varieties.

We give a sufficient condition for a Brauer-Severi surface bundle over a rational 3-fold to not be

stably rational. Additionally, we present an example that satisfies this condition and demonstrate

the existence of families of Brauer-Severi surface bundles whose general members are smooth and

not stably rational.

Copyright by
SHITAN XU
2025

ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor, Dr. Rajesh Kulkarni, for all of his guidance,

friendship and support throughout my Ph.D studies. I also thank him for introducing me to the

realm of algebraic geometry and offering me such an interesting topic. This thesis would not have

been possible without his endless patience and unwavering support. I am profoundly grateful for

his efficiency and mathematical rigor within our countless conversations.

I also wish to express my sincere appreciation to the members of my dissertation committee:

Dr. Aaron Levin, Dr.

Igor Rapinchuk, Dr. Michael Shapiro and Dr. Linhui Shen, for their

exceptional guidance and dedication in mentoring me over the years. I also extend my gratitude

to Prof. Pirutka, Prof. Auel, Prof.Kresch and Prof. Schreieder for their comments and suggestions

during my research process.

I would like to thank my academic siblings, without whom I would not have made it through my

PhD degree: Charlotte Ure, Nicholas Rekuski, Michael Annunziata, Yizhen Zhao and Yu Shen. I

will always keep in mind that how our math insights and friendship grows together.

Thank you to my parents: Mr. Shufeng Xu and Mrs. Yujie Song, for your steady support

through my whole life. To all of my family members, especially my grandparents: Mr. Yukun Xu,

Mr. Dengshun Song, Mrs. Xiucheng Wang, Mrs.Yahua Liang, for providing me a warm childhood.

It is impossible to fully grasp how much you sacrificed to raise me until I began to see you as a real

person rather than just a role model.

To my love, Jinting Liang, Thank you for your emotional support and companionship. Looking

back, every laugh, every tear, and every word have become cherished memories. I will always

remember the vibrant and spirited times we shared and embrace the even brighter future ahead of

us.

Working in the Math Department at Michigan State University has been a truly remarkable

journey! I am deeply grateful to everyone who has been part of this experience. Your support and

influence have been priceless.

iv

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

TABLE OF CONTENTS

CHAPTER 2

2.1 Brauer groups .
2.2 Cyclic algebras
2.3 Unramified Brauer group and purity . . . . . . . . . . . . . . . . . . . . . .

.
BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1

6
6
8
9

CHAPTER 3

BRAUER-SEVERI SURFACE BUNDLES . . . . . . . . . . . . . . . . 11
3.1 Review of conic bundles
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Discriminant locus
3.3 Brauer groups of Brauer-Severi surface bundles . . . . . . . . . . . . . . . . . 16

.

CHAPTER 4

FLATNESS AND EXAMPLES . . . . . . . . . . . . . . . . . . . . . . 25
4.1 A distinguished example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
. 29
4.2 Flatness .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

. .

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.

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CHAPTER 5

THE SPECIALIZATION METHOD AND DESINGULARIZATION .

. 35
5.1 The specialization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Desingularization .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
. .
5.3 Main result

. .

.
.

BIBLIOGRAPHY . .

. .

APPENDIX

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

v

CHAPTER 1

INTRODUCTION

Rationality questions aim to determine the parametrization of multivariable polynomials. To avoid

unnecessary complexity, we focus only on parametrizations involving polynomials or rational func-

tions (quotients of polynomials). Such a parametrization is commonly referred to as a birational

parametrization. A first example would be the parametrization of the unit circle 𝑥2 + 𝑦2 − 1 = 0.

Example 1.0.1. 𝑥 = 𝑐𝑜𝑠(𝜃); 𝑦 = 𝑠𝑖𝑛(𝜃) is not a birational parametrization of the unit circle.
1+𝑡2 ; 𝑦 = 1−𝑡2
Because these expressions involve power series instead of polynomials. However, 𝑥 = −2𝑡
1+𝑡2

is a birational parametrization of unit circle.

Given a complex projective integral variety 𝑋. We say 𝑋 is rational if it admits such a birational

parametrization. A formal definition is usually given from the geometric point of view:

Definition 1.0.2. Let 𝑋 be a projective integral variety over C. We say that 𝑋 is rational if 𝑋 is

birational to a projective space P𝑛

𝑘 for some natural number 𝑛. 𝑋 is stably rational if there exists a

natural number 𝑚 such that 𝑋 ×𝑘 P𝑚

𝑘 is rational.

It is clear from the definition that rational varieties are stably rational. However the converse

is false, the first example is given in [BCTSSD85]. Their example admits a conic bundle structure

over a rational surface. In [HPT18], an example of a quadratic surface bundle over P2 that is not

stably rational and has a nontrivial unramified Brauer group is constructed. This example is realized

as a divisor in P2 × P3 of bidegree (2,2), with the quadratic surface bundle structure induced by the

first projection. In [ABvBP20], this elegant example was examined from a different perspective:

it naturally becomes a conic bundle over P3 via the second projection. This structure allowed the

authors to establish a sufficient condition [ABvBP20, Thm. 2.6] for a conic bundle over P3 to not

be stably rational.

All of the above examples illustrate a slogan: It is easier to determine rationality of varieties

admitting a fibration structure. Why is this true? Because there is a very powerful tool to show that

a variety is not stably rational, the unramified Brauer group.

1

Let 𝑋 be a projective integral variety over 𝐶. By definition, the unramified Brauer group of 𝑋

is a specific subgroup of the Brauer group of the function field of 𝑋, which is a stably birational

invariant. When 𝑋 admits a fibration structure over a base, we can analyze which Brauer classes

originate from the base. This approach can lead to the construction of a nontrivial unramified

Brauer class for 𝑋, which is sufficient to show that 𝑋 is not stably rational.

In practice, we usually allow 𝑋 to be singular. The specialization method, introduced by

Voisin in [Voi15] and further developed by Colliot-Thélène and Pirutka in [CTP16], as well as by

Schreieder in [Sch19a, Proposition 26], gives us a way to get smooth non-stably rational varieties

from a singular one. For example, the authors in [ABvBP20] also introduced a new example of a

flat family of conic bundles over P3, where a very general member is not stably rational, using a

theorem of Voisin [Voi15, Thm. 2.1], in the form of [CTP16, Thm. 2.3].

On the other hand, to apply this technic, we require the singular locus of 𝑋 is mild. To be more

precise, we would like 𝑋 admits a desingularization which is universally CH0-trivial:

Definition 1.0.3. A universally CH0-trivial desingularization of 𝑋 is a projective morphism

𝑓 : ˜𝑋 → 𝑋, such that ˜𝑋 is smooth and for any field extension 𝐹/C, the map induced by 𝑓 :

CH0( ˜𝑋𝐹) → CH0(𝑋)

is an isomorphism. Here CH0(𝑋) denotes the 0-th Chow group, that is the group of 0-cycles

module rational equivalence [Ful98, Chapter 1].

In a summary, to apply this specialization method, there are two requirements:

• Constructing a specific example with meaningful, nontrivial stably birational invariants that

can be specialized, such as the unramified Brauer group.

• Showing that this example admits a universally CH0-trivial desingularization and can spe-

cialize to varieties of interest.

However, it is usually hard to construct an explicit desingularization of a given variety. Fur-

thermore, it is also hard to check the universally CH0-trivial property using Definition 1.0.3. A

2

good refinement to check this property is given in [CTP16, Proposition 1.8], but one still need to

construct an explicit smooth model of the given variety. Recently, Schreieder gave an alternate

approach in a series of papers: [Sch19a, Proposition 26] and [Sch19b]. Instead of constructing

such a desingularization, Schreieder’s result allows a purely cohomological criteria.

In a recent paper [Pir23], Pirutka introduced the notion of the relative unramified cohomol-

ogy group, which combines the approach of constructing nontrivial unramified Brauer class via

fibrations ([AM72],[CTO89]) and the method given in [Sch19a] and [Sch19b] to avoid geometric

construction of universally 𝐶𝐻0-trivial desingularization.

The present thesis is going to study stable rationality of Brauer-Severi surface bundles over

rational 3-folds using the specialization method mentioned above. We start with the definitions:

Definition 1.0.4 ([GS06, Def. 5.1.1]). A Brauer-Severi variety of dimension 𝑛 over a field 𝑘 is a

projective algebraic variety 𝑋 over 𝑘 such that the base extension 𝑋 ¯𝑘 := 𝑋 ×𝑘 ¯𝑘 becomes isomorphic
to P𝑛

¯𝑘 , where ¯𝑘 is an algebraic closure of 𝑘.

Remark 1.0.5.

1. Projective spaces are considered as trivial Brauer-Severi varieties.

2. 1-dimensional Brauer-Severi varieties are precisely smooth projective plane conics ([GS06,

Chpater 5]).

3. Considering 2, a conic bundle, which by definition is a flat projective surjective morphism of

varieties with geometric fibers isomorphic to projective plane conics and general fiber smooth,

is a synonym for 1-dimensional Brauer-Severi bundle or Brauer-Severi curve bundle.

In this thesis, we mainly focus on the structure of 2-dimensional Brauer-Severi bundle:

Definition 1.0.6 ([KT19, Def. 4.1]). Let 𝐵 be a locally Noetherian scheme, in which 3 is invertible

in the local rings. A Brauer-Severi surface bundle over 𝐵 is a flat projective morphism 𝜋 : 𝑌 → 𝐵

such that the fiber over every geometric point of 𝐵 is isomorphic to one of the following:

• P2

3

• The union of three standard Hirzebruch surfaces F1, meeting transversally, such that any pair

of them meets along a fiber of one and the (−1)-curve of the other.

• An irreducible scheme whose underlying reduced subscheme is isomorphic to the cone over

a twisted cubic curve.

Remark 1.0.7. In the case of conic bundles, the degenerate fibers are simply two distinct lines or

a double line. The 2 dimensional case is more complex.

Kresch and Tschinkel introduced good models of Brauer-Severi surface bundles using the

concept of a root stack in [KT19]. With this definition, they constructed a flat family of Brauer-

Severi surface bundles over P2 [KT20, Thm. 1], in which the general member is smooth and

not stably rational. A natural next step following these advancements is to investigate the stable

rationality of Brauer-Severi surface bundles over P3.

This paper begins by generalizing Theorem 2.6 of [ABvBP20] to Brauer-Severi surface bundles

over 3-folds. After constructing a new (singular) example with a nontrivial unramified Brauer

group, we obtain the following result:

Theorem 1.0.8. There exists a flat projective family of Brauer-Severi surface bundles over P3
C,

where a general fiber in this family is smooth and not stably rational.

The structure of this thesis is as follows: In Chapter 2, we review basic facts about Brauer

groups (Section 2.1, Section 2.2) and introduce the unramified Brauer group (Section 2.3), a stably

birational invariant. By definition, this is the subgroup of the Brauer group of the function field,

whose elements arise from the Brauer classes of a smooth model. Recently, significant progress

has been made using the unramified Brauer group and the specialization method to show that a

very general member of certain classes of varieties is not stably rational ([Pir18, Section 2.1]).

In Chapter 3, after recalling a criterion for stable rationality of conic bundles over 3-folds

([ABvBP20, Thm. 2.6]) in Section 3.1. We give the restrictions of discriminant locus of Brauer-

Severi surface bundles in Section 3.2. Then we provide the key technical theorem in Section 3.3.

4

This provides a tool to construct an explicit (singular) Brauer-Severi surface bundle over P3

C with

a nontrivial unramified Brauer group, which we explore in Chapter 4. Section 4.1 consists of

explicit construction of our example, and in Section 4.2, we verify that this example is indeed a

Brauer-Severi surface bundle.

Chapter 5 it ought to verify the example we constructed can be used as a reference variety. We

recall the main developments of specialization methods dated back to 70s until now in Section 5.1.

In Section 5.2, we show that our example 4.1.1 satisfies the hypotheses required by the specialization

method introduced by Voisin in 2014 [Voi15], and further developed by Colliot-Thélène and Pirutka

in 2016 [CTP16]. Following Schreieder’s approach [Sch19a, Proposition 26], we verify this using

a purely cohomological criterion. In Section 5.3, we prove Theorem 1.0.8 by constructing a flat

family of Brauer-Severi surface bundles over P3

C, where the general member is smooth and includes

Example 4.1.1 as a member. Detailed calculations are provided separately in Appendix.

5

CHAPTER 2

BACKGROUND

In this chapter, we recall the necessary background that will be used later.

2.1 Brauer groups

For a more detailed treatment of Brauer groups of fields and schemes, see [GS06] and [CTS21].

let 𝑘 be a field, 𝐴 be an associative 𝑘-algebra. 𝐴 is called central if its center is 𝑘. 𝐴 is called simple

if it has no non-trivial two-sided ideals. A central simple k-algebra(CSA) is a finite dimensional

𝑘-algebra that is both central and simple.

Example 2.1.1. Here are first examples of central simple algebras:

• Hamilton quaternions over R. Generally, any central division algebra is a CSA.

• The 𝑛 × 𝑛 matrix algebra 𝑀𝑛 (𝑘), more generally, if 𝐷 is a central division algebra, then

𝑀𝑛 (𝐷) is a CSA.

Wedderburn’s theorem provides a converse to the second type of examples above:

Theorem 2.1.2 (Wedderburn,[GS06, Thm. 2.1.3]). For any CSA 𝐴, there is a central division

algebra 𝐷 and an integer 𝑛 ≥ 1, such that

𝐴 (cid:27) 𝐷 ⊗𝑘 𝑀𝑛 (𝑘) = 𝑀𝑛 (𝐷)

And the division algebra 𝐷 is unique up to isomorphism.

If 𝐴 is a CSA over 𝑘 and 𝐾/𝑘 is a field extension such that 𝐴 ⊗𝑘 𝐾 (cid:27) 𝑀𝑛 (𝐾) for suitable 𝑛.

Then 𝐾 is called a splitting field of 𝐴. Splitting field of a given CSA always exists according to

the definition. In fact, we can do better:

Theorem 2.1.3 ([GS06, Thm. 2.2.1]). Let 𝑘 be a field and let 𝐴 be a finite-dimensional 𝑘-algebra.

Then 𝐴 is a CSA if and only if there is a positive intger 𝑛 and a finite field extension 𝐾/𝑘 such that

𝐴 ⊗𝑘 𝐾 (cid:27) 𝑀𝑛 (𝐾)

6

As a direct corollary of Theorem 2.1.3, we know the dimension of a given CSA as a 𝑘-algebra

is a square, We call the integer

√

𝑑𝑖𝑚𝑘 𝐴 degree of 𝐴. By Wedderburn’s Theorem 2.1.2, assume

𝐴 (cid:27) 𝑀𝑛 (𝐷), we define index of 𝐴 to be the degree of 𝐷.

Definition 2.1.4. Two CSA 𝐴 and 𝐵 are called Brauer equivalent if there are 𝑛, 𝑚 > 0 such that

𝐴 ⊗𝑘 𝑀𝑛 (𝑘) (cid:27) 𝐵 ⊗𝑘 𝑀𝑚 (𝑘)

This is an equivalent relation on the set of all central simple algebras over 𝑘. Given a CSA 𝐴, use

[ 𝐴] denote its Brauer equivalence class.

Definition 2.1.5. The Brauer group of a field 𝑘, denote by Br(𝑘), is the set of Brauer equivalent

classes of central simple algebras over 𝑘, with a structure of torsion abelian group with the binary

operation given by:

[ 𝐴] + [𝐵] = [ 𝐴 ⊗𝑘 𝐵]

The index of a Brauer class [ 𝐴] is defined to be the index of 𝐴, and the order [ 𝐴] in Br(𝑘) is called

the period of [ 𝐴].

Example 2.1.6.

• Br(𝑘) = 0, if 𝑘 is separably closed or is finite.

• (Frobenius) Br(R) (cid:27) Z/2Z.

• (Tsen) let 𝑘 be a field of transcendence degree 1 over an algebraically closed field, then

Br(𝑘) = 0.

We can also describe the Brauer group using Galois cohomology:

Theorem 2.1.7 ([GS06, Thm. 4.4.7]). Let 𝑚 be a positive integer that is invertible in 𝑘, then

Br(𝑘) (cid:27) 𝐻2(𝑘, G𝑚), Br(𝑘) [𝑚] (cid:27) 𝐻2(𝑘, 𝜇𝑚)

Situations are become complex when we passes to schemes, there are naturally two ways

to define a Brauer group of a scheme: namely following the idea of Definition 2.1.5 or using

cohomological description similar to Theorem 2.1.7. However, the two constructions are not

7

equivalent. For details in this direction, see [CTS21, Chapter 3,4]. In this thesis, when talking

about Brauer group of schemes, we adopt the following definition:

Definition 2.1.8. Let 𝑋 be a quasi-compact 𝑘-scheme, the (cohomological) Brauer group, denote

by Br(𝑋), is defined to be the torsion subgroup of 𝐻2

é𝑡 (𝑋, G𝑚). In particular, assume 𝑚 is a positive

integer that is invertible in 𝑘, the 𝑚-tosion part Brauer group of 𝑋 is Br(𝑋) [𝑚] = 𝐻2

é𝑡 (𝑋, 𝜇𝑚).

2.2 Cyclic algebras

Through out this section, we assume the base field 𝑘 contains all primitive roots of unity. 𝑝 is

a prime number.

Definition 2.2.1. For any 𝑎, 𝑏 ∈ 𝑘 ∗, and a primitive 𝑝-th root of unity 𝜔 ∈ 𝑘 ∗, we define the cyclic

algebra, denote by (𝑎, 𝑏)𝜔, to be the 𝑘-algebra by the following presentation:

(𝑎, 𝑏)𝜔 = ⟨𝑥, 𝑦|𝑥 𝑝 = 𝑎, 𝑦 𝑝 = 𝑏, 𝑥𝑦 = 𝜔𝑦𝑥⟩ ,

Remark 2.2.2.

• when 𝑝 = 2, 𝜔 = −1, we get back the definition of quaternion algebras.

• One directly check (𝑎, 𝑏)𝜔 is a CSA over 𝑘. Futhermore, its either a division algebra of

degree (index) p, or the matrix algebra 𝑀𝑝 (𝑘). The later case happens if and only if ,by
[GS06, Corollary 4.7.7], 𝑏 is a norm from the field extension 𝑘 ( 𝑚√

𝑎)/𝑘.

The importance of cyclic algebras is that they served as nice representatives of Brauer classes,

as a result of Merkurjev-Suslin theorem:

Theorem 2.2.3 ((Merkurjev-Suslin),[GS06, Thm. 8.6.5]). With the above assumptions on 𝑘 and 𝑝,

let 𝛼 ∈ Br(𝑘) [ 𝑝], then there exist a finite number of cyclic algebras of degree p {(𝑎𝑖, 𝑏𝑖)}𝑡

𝑖=1, such

that

𝛼 = [(𝑎1, 𝑏1)𝜔 ⊗𝑘 · · · ⊗𝑘 (𝑎𝑡, 𝑏𝑡)𝜔]

In the rest of the thesis, we will focus on cyclic algebras of degree 3. Note that when 𝑘 contains a

primitive cubic roots of unity, we have a non-canonical isomorphism of group schemes :𝜇3 (cid:27) Z/3Z,

8

hence the 3-torsion of the Brauer group of a field k would be

Br(𝑘) [3] (cid:27) 𝐻2(𝑘, Z/3Z)

2.3 Unramified Brauer group and purity

Let 𝐿 be the function field of an integral scheme 𝑍 over the field of complex numbers, C. Let

𝜈 be a discrete valuation of 𝐿 with residue field 𝑘 (𝜈), we have the following residue maps [GS06,

Section 6.8]:

𝜈 : 𝐻1(𝐿, Z/3) → 𝐻0(𝑘 (𝜈), Z/3)
𝜕1

𝜈 : 𝐻2(𝐿, Z/3) → 𝐻1(𝑘 (𝜈), Z/3)
𝜕2

By Kummer theory [GS06, Prop. 4.3.6], we can identify these residue maps as:

𝜈 : 𝐿×/𝐿×3 → Z/3
𝜕1

𝜈 : Br(𝐿) [3] → 𝑘 (𝜈)×/𝑘 (𝜈)×3
𝜕2

Lemma 2.3.1. With notations as above, the two residue maps are defined by:

𝜈 ( [𝑎]) = 𝜈(𝑎) (mod 3)
𝜕1

𝜈 ([(𝑎, 𝑏)𝜔]) = (−1)𝜈(𝑎)𝜈(𝑏) 𝑎𝜈(𝑏)

𝜕2

𝑏𝜈(𝑎) mod 𝑘 (𝜈)×3

Proof. See [IOOV17, Thm. 2.18]

□

Definition 2.3.2. The 3-torsion of unramified Brauer group of a field 𝐿 over another field 𝑘,

denoted by 𝐻2

𝑛𝑟 (𝐿/𝑘, Z/3Z) or Br𝑛𝑟 (𝐿/𝑘) [3], is the intersection of kernels of all residue maps 𝜕2
𝜈 ,

where 𝜈 take values in all divisorial valuations of 𝐿 which are trivial on 𝑘.

From the discussions in [CT95] and [CTS21, Corollary 6.2.10], the unramified Brauer group

of L is a stably birational invariant for any model X of L, whether the model is nonsingular or

singular. Here, a model X of L refers to an integral projective variety with function field L. If the

model is nonsingular, we have the following:

9

Lemma 2.3.3. Let 𝑘 be a field with characteristic not 3. Let 𝑋 be a regular, proper, integral variety

over 𝑘 with function field 𝐿, then we have

Br(𝑋) [3] (cid:27) Br𝑛𝑟 (𝐿/𝑘) [3]

Proof. This is a direct result of [CTS21, Prop. 3.7.8]

□

Given a 3-torsion Brauer class in the Brauer group of the function field L, it is not easy to

determine whether it belongs to the unramified subgroup using the definition alone, as there are

usually too many divisorial valuations to consider. In [CT95], several theorems are established that

reduce the number of valuations needed to check whether a Brauer class is unramified. Specifically,

it suffices to check valuations corresponding to prime divisors in a smooth model. This result follows

from the purity property of unramified Brauer groups. For more details on these theorems, see

[CT95] and [CTS21, Section 3.7]. A similar discussion can be found in [ABvBP20, Section 2].

10

CHAPTER 3

BRAUER-SEVERI SURFACE BUNDLES

3.1 Review of conic bundles

In this section, we make a brief review of a formula for the unramified Brauer group of centain

conic bundle fourfolds, all of the contents in this section can be found in [ABvBP20]. We start

with the definition:

Definition 3.1.1. Let 𝐵 be a Noetherian scheme over C, a conic bundle over 𝐵 is a flat projective

morphism 𝜋 : 𝑌 → 𝐵 such that the fiber over every geometric point of 𝐵 is isomorphic to one of

the following:

• P1

• Two distinct lines intersect at one point

• Double lines

In [ABvBP20], The authors focused on conic bundles over P3

C be such a conic
C be its discriminant locus, which by definition is the subset of points in P3
C
whose fibers are not smooth plane conics. One can show that 𝑆 is indeed a divisor of pure dimension

bundle, and let 𝑆 ⊂ P3

C. Let𝜋 : 𝑌 → P3

with its natural determinantal scheme structure, and since P3

C is Noetherian, 𝑆 has finitely many

irreducible components, denoted by 𝑆1, · · · 𝑆𝑛.

Definition 3.1.2 ([ABvBP20, Def. 2.4]). The discriminant locus is good if 𝑆 is reduced and for

each irreducible component 𝑆𝑖, the fiber 𝑌𝑠 for general 𝑠 ∈ 𝑆𝑖 consists of two distinct lines, and the

natural double cover of 𝑆𝑖 induced by 𝜋 is irreducible.

With the above requirements of discriminant locus of conic bundles, one can prove the following

formula for the unramified Brauer group:

Theorem 3.1.1 ([ABvBP20, Thm. 2.6]). Let 𝑘 be an algebraically closed field of characteristic

not 2 and let 𝜋 : 𝑌 → 𝐵 be a conic bundle over a smooth projective threefold 𝐵 over 𝑘. Assume

11

Br(𝐵) [2] = 0, 𝐻3
ét

(𝐵, Z/2) = 0, (e.g. 𝐵 = P3). Let 𝛼 ∈ Br(𝐾) [2] be the Brauer class in 𝐾 = 𝑘 (𝐵)

corresponding to the generic fiber of 𝜋. Assume the discriminant locus is good (Definition 3.1.2)

with components 𝑆1, · · · , 𝑆𝑛. And also assume that

1)Through any irreducible curve in 𝐵, there pass at most two surfaces from the set 𝑆1, · · · , 𝑆𝑛.

2) Through any point of 𝐵, there pass at most three surfaces from the set 𝑆1, · · · , 𝑆𝑛.

3)For all 𝑖 ≠ 𝑗, 𝑆𝑖 and 𝑆 𝑗 are factorial at every point of 𝑆𝑖 ∩ 𝑆 𝑗 .

Let

𝛾𝑖 = 𝜕2

𝑆𝑖 (𝛼) ∈ 𝐻1(𝑘 (𝑆𝑖), Z/2)

Define a subgroup Γ of the group (cid:201)𝑛
𝑖=1

𝐻1(𝑘 (𝑆𝑖), Z/2) by

Γ =

𝑛
(cid:202)

𝑖=1

< 𝛾𝑖 >(cid:27) (Z/2)𝑛

We will write elements of Γ as (𝑥1, 𝑥2 · · · , 𝑥𝑛) with 𝑥𝑖 ∈ {0, 1}.

Let 𝐻 ⊂ Γ consists of those elements (𝑥1, · · · , 𝑥𝑛) ∈ (Z/2)𝑛 such that 𝑥𝑖 = 𝑥 𝑗 whenever there

exists an irreducible components 𝐶 of 𝑆𝑖 ∩ 𝑆 𝑗 , such that

(𝜕2

𝑆𝑖 (𝛼), 𝜕2

𝑆 𝑗 (𝛼)) = (0, 0)or(1, 1)

Let 𝐻′ ⊂ 𝐻 consists of those elements (𝑥1, · · · , 𝑥𝑛) ∈ (Z/2)𝑛 such that 𝑥𝑖 = 𝑥 𝑗 whenever there

exists an irreducible components 𝐶 of 𝑆𝑖 ∩ 𝑆 𝑗 , such that

(𝜕2

𝑆𝑖 (𝛼), 𝜕2

𝑆 𝑗 (𝛼)) = (0, 0) and 𝛾𝑖 |𝐶 and 𝛾 𝑗 |𝐶 are not both zero in 𝐻1(𝑘 (𝐶), Z/2)

Then 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑘, Z/2) contains the subquotient 𝐻′/< 1, · · · , 1 >.

These theorem applies to a famous example given in [HPT18]: Let 𝑌𝐻𝑇 𝑃 denote the divisor of

bi-degree (2, 2) in P2

C × P3

C given by :

𝑌 𝑍 𝑆2 + 𝑋 𝑍𝑇 2 + 𝑋𝑌𝑈2 + (𝑋 2 + 𝑌 2 + 𝑍 2 − 2(𝑋𝑌 + 𝑋 𝑍 + 𝑌 𝑍))𝑉 2 = 0

where we assume the coordinates in P3

C are [𝑆 : 𝑇 : 𝑈 : 𝑉] and the coordinates in P2

C are [𝑋 : 𝑌 : 𝑍].

By considering the projection from 𝑌𝐻𝑇 𝑃 to P2

C, the authors in [HPT18] proved this is not stably

12

rational as a quadratic surface bundle. In [ABvBP20], the authors considering the projection from

𝑌𝐻𝑇 𝑃 to P3

C instead, and Theorem 3.1.1 double checked that 𝑌𝐻𝑇 𝑃 is not stably rational using its

conic bundle structure over P2
C.

3.2 Discriminant locus

Now we generlize the results from previous sections to the case of Brauer-Severi surface bundles.

The first step is to find suitable discriminant locus.

Let 𝜋 : 𝑌 → 𝐵 be a Brauer-Severi surface bundle (Definition 1.0.6) over a smooth projective ra-

tional threefold 𝐵 over a field 𝑘 whose generic fiber is smooth, and let 𝑆 = {𝑏 ∈ 𝐵|𝜋−1(𝑏) is singular}

denote its discriminant locus. We consider 𝑆 with its reduced closed subscheme structure in 𝐵, and

since 𝐵 is Noetherian, 𝑆 consists of finitely many irreducible components, say 𝑆1, · · · , 𝑆𝑛.

In the case of conic bundles considered in [ABvBP20, Def. 2.4], the authors focused on a

special kind of conic bundles with a good discriminant locus. We will generalize the definition of

a good discriminant locus to Brauer-Severi surface bundles in this context:

Definition 3.2.1. We say the discriminant locus 𝑆 is good if the following conditions are satisfied:

1. Each irreducible component of 𝑆 is reduced. (This is assumed above.)

2. The fiber 𝑌𝑠 over a general 𝑠 ∈ 𝑆𝑖 for each irreducible component 𝑆𝑖 is geometrically the

union of three standard Hirzebruch surfaces F1 described in Definition 1.0.6.

3. The natural triple cover of 𝑆𝑖 induced by 𝜋 : 𝑌 → 𝐵 is irreducible.

4. By (3), the fiber 𝑌𝐹𝑆𝑖 over the generic point of 𝑆𝑖 is irreducible. Thus, there is a natural map
𝜏 from the cubic classes of function field of 𝑆𝑖 to the cubic classes of function field of 𝑌𝐹𝑆𝑖 .
We assume the cubic extension over the function field of 𝑆𝑖 induced by 𝜋 : 𝑌 → 𝐵 generates

the kernel of 𝜏.

Remark 3.2.2. The last requirement is automatically satisfied in the case of conic bundles with the

suitable definition given in [ABvBP20, Thm. 2.6]. However, in our case, this is not generally true.

Note that the example we provided meets all these requirements (See Example 4.1.1).

13

Remark 3.2.3. By Lemma 2.3.1, those 𝑆𝑖 are precisely those irreducible surfaces such that the

Brauer class of the generic fiber have a nontrivial residue along 𝑆𝑖. In particular, the discriminant

locus is a divisor of the base with pure-dimensional irreducible components.

We end this section with a lemma that generalizes [ABvBP20, Lemma 2.3] to the 3-torsion

case:

Lemma 3.2.4. Let 𝑆 be a smooth nonsplit Brauer-Severi surface over an arbitrary field 𝐾 (in

particular, 𝑆 ¯𝐾 (cid:27) P2

¯𝐾). Then the pullback map Br(𝐾) → Br(𝑆) induces an exact sequence:

0 → Z/3 → Br(𝐾) → Br(𝑆) → 0,

where the kernel is generated by the Brauer class 𝛼 ∈ Br(𝐾) [3] determined by 𝑆. Furthermore, if

characteristic of 𝐾 ≠ 3 and 𝐾 contains a primitive 9-th root of unit, then the above exact sequence

restricts to :

0 → Z/3 → Br(𝐾) [3] → Br(𝑆) [3] → Z/3 → 0

Proof. We first prove the exactness of the first sequence. Recall that the kernel ker(Br(𝐾) → Br(𝑆))

is given by Amitsur’s theorem [GS06, Thm. 5.4.1]. To show surjectivity of Br(𝐾) → Br(𝑆),

consider the separable closure 𝐾 𝑠 of 𝐾, and let Γ = Gal(𝐾 𝑠/𝐾). Then we have the Hochschild-

Serre spectral sequence

𝐻 𝑝 (Γ, 𝐻𝑞 (𝑆𝐾 𝑠 , G𝑚)) ⇒ 𝐻 𝑝+𝑞 (𝑆, G𝑚).

The low degree exact sequence reads

0 → Pic(𝑆) → Pic(𝑆𝐾 𝑠 )Γ → Br(𝐾) → ker

(cid:16)

Br(𝑆) → Br(𝑆𝐾 𝑠 )Γ(cid:17)

→ 𝐻1(Γ, Z)

By the definition of a Brauer-Severi Variety, we know 𝑆𝐾 𝑠 (cid:27) P2

𝐾 𝑠 . Hence we have Br(𝑆𝐾 𝑠 ) (cid:27)

Br(𝐾 𝑠) = 0. Since we clearly have 𝐻1(Γ, Z) = 0, it follows that Br(𝐾) → Br(𝑆) is surjective.

For the second part, consider any 𝑎 ∈ Br(𝑆) [3]. There exists a lift 𝑎′ ∈ Br(𝐾) of 𝑎 such that

3𝑎′ ↦→ 0 ∈ Br(𝑆). Therefore, 3𝑎′ ∈ ker (Br(𝐾) → Br(𝑆)) (cid:27) Z/3. In other words, 9𝑎′ = 0 ∈

14

Br(𝐾). Hence, Br(𝐾) [9] surjects onto Br(𝑆) [3]. Notice that by the description of the kernel in

Amitsur’s theorem, we clearly have

ker (Br(𝐾) [3] → Br(𝑆) [3]) (cid:27) Z/3.

To compute the cokernel, consider the short exact sequence of trivial Γ-modules:

1 → 𝜇3 → 𝜇9 → 𝜇3 → 1,

and its associated long exact sequence:

· · · → 𝐻1(𝐾, 𝜇3)

𝜕1
−−→ 𝐻2(𝐾, 𝜇3) → 𝐻2(𝐾, 𝜇9) → 𝐻2(𝐾, 𝜇3)

𝜕2
−−→ 𝐻3(𝐾, 𝜇3) → · · ·

We claim the boundary maps 𝜕1, 𝜕2 in the above exact sequence are zero.

Indeed, let 𝜔 be a

primitive third root of unit. By the proof of [TOP17, Lemma 4.1], 𝜕1 is given by the cup product

with [𝜔] ∈ 𝐻1(𝐾, Z/3). By our assumption, 𝜔 is a cube in 𝐾, hence [𝜔] = 0 ∈ 𝐻1(𝐾, Z/3) (cid:27)

𝐾 ∗/(𝐾 ∗)3. This shows 𝜕1 = 0.

To show 𝜕2 = 0, consider the following commutative diagram given in [GS06, Lemma 7.5.10]:

𝜇3 ⊗ 𝐾 𝑀

2 (𝐾)

𝜔∪ℎ2

𝐾 ,3

𝐻2(𝐾, 𝜇⊗3
3

)

{,}

˜𝜕2

3 (𝐾)/3𝐾 𝑀
𝐾 𝑀

3 (𝐾)

ℎ3

𝐾 ,3

𝐻3(𝐾, 𝜇⊗3
3

)

Notations in the above diagram are explained in [GS06, Lemma 7.5.10]. Notice since 𝜇3 are trivial

Γ-modules, we have the following isomorphism ([TOP17, Lemma 4.1]):

𝑗 : 𝐻𝑖 (𝐾, Z/3) (cid:27) 𝐻𝑖 (𝐾, 𝜇⊗ 𝑗
𝜙𝑖

3

)

𝛼 ↦→ 𝛼 ∪ (𝜔 ∪ · · · ∪ 𝜔)
(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)
(cid:125)
(cid:124)

(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)

(cid:123)(cid:122)
𝑗 copies

Since the upper horizontal map in the commutative diagram is given by the symbol product

with 0 = [𝜔] ∈ 𝐻1(𝐾, Z/3), it follows that this map is 0. By the Merkurjev-Suslin theorem
([GS06, Theorem 8.6.5]), the left vertical map is surjective, hence ˜𝜕2 = 0. Given that cup

15

products "commute" with boundary homomorphisms ([NSW08, Proposition 1.4.3]), we have the

commutative diagram involving 𝜕2:

𝐻2(𝐾, 𝜇⊗3
3

)

(𝜙2
3

) −1

𝐻2(𝐾, Z/3)

˜𝜕2

𝜕2

𝐻3(𝐾, 𝜇⊗3
3

)

(𝜙3
3

) −1

𝐻3(𝐾, Z/3)

From this, it is clear that 𝜕2 = 0.

Next notice that 𝐾 has characteristic ≠ 3, so we have Br(𝐾) [𝑛] (cid:27) 𝐻2(𝐾, 𝜇𝑛) when 𝑛 is a power

of 3. Consider the following commutative diagram:

0

0

Br(𝐾) [3]

Br(𝐾) [9]

Br(𝐾) [3]

0

𝜎3

𝜎9

𝜎3

Br(𝑆) [3]

Br(𝑆) [9]

Br(𝑆) [3]

Then the snake lemma gives us

ker(𝜎9)

𝜙1−−→ ker(𝜎3) → coker(𝜎3)

𝜙2−−→ coker(𝜎9).

Since 𝜙1 is multiplication by 3 and ker(𝜎9) (cid:27) Z/3, 𝜙1 = 0. The map 𝜙2 is also zero as Br(𝐾) [9]

maps onto Br(𝑆) [3]. Hence we have coker(𝜎3) (cid:27) ker(𝜎3) (cid:27) Z/3.

□

3.3 Brauer groups of Brauer-Severi surface bundles

As mentioned in the introduction, we interest in Brauer-Severi surface bundles over P3

C. In this

section, we present sufficient conditions under which a Brauer-Severi surface bundle 5-fold is not

stably rational. These conditions are derived by generalizing [ABvBP20, Thm. 2.6] to the 3-torsion

case. However, the details in these two cases are quite different.

Theorem 3.3.1. Let 𝑘 be an algebraically closed field of characteristic ≠ 3 and let 𝜋 : 𝑌 → 𝐵 be

a Brauer-Severi surface bundle over a smooth projective threefold 𝐵 over 𝑘 with a smooth generic

fiber. Assume Br(𝐵) [3] = 0 and 𝐻3
ét

(𝐵, Z/3) = 0. (For example, take 𝐵 = P3.) Let 𝛼 ∈ Br(𝐾) [3]

be the Brauer class over 𝐾 = 𝑘 (𝐵) corresponding to the generic fiber of 𝜋, and it can be represented

by a cyclic algebra of index 3. Assume the discriminant locus is good [Definition 3.2.1] with

irreducible components 𝑆1, · · · , 𝑆𝑛. Further suppose the following conditions also hold:

16

1. Any irreducible curve in 𝐵 is contained in at most two surfaces from the set {𝑆1, · · · , 𝑆𝑛}.

2. Through any point of 𝐵, there pass at most three surfaces from the set {𝑆1, · · · , 𝑆𝑛}.

3. For all 𝑖 ≠ 𝑗, 𝑆𝑖 and 𝑆 𝑗 are factorial at every point of 𝑆𝑖 ∩ 𝑆 𝑗 .

Let 𝛾𝑖 = 𝜕2
𝑆𝑖

(𝛼) ∈ 𝐻1(𝑘 (𝑆𝑖), Z/3). Let Γ be the subgroup of (cid:201)𝑛
𝑖=1

𝐻1(𝑘 (𝑆𝑖), Z/3) given by

Γ = (cid:201)𝑛

𝑖=1 ⟨𝛾𝑖⟩ (cid:27) (Z/3)𝑛. We write elements of Γ as (𝑥1, 𝑥2 · · · , 𝑥𝑛) with 𝑥𝑖 ∈ {0, 1, 2}.

Let 𝐻 ⊂ Γ consist of those elements (𝑥1, · · · , 𝑥𝑛) ∈ (Z/3)𝑛 such that 𝑥𝑖 = 𝑥 𝑗 whenever there

exists an irreducible component 𝐶 of 𝑆𝑖 ∩ 𝑆 𝑗 , such that

(𝜕1

𝐶 (𝛾𝑖), 𝜕1

𝐶 (𝛾 𝑗 )) = (1, 2) or (2, 1).

Let 𝐻′ ⊂ 𝐻 be the subgroup consisting of elements (𝑥1, · · · , 𝑥𝑛) ∈ (Z/3)𝑛 such that 𝑥𝑖 = 𝑥 𝑗
whenever there exists an irreducible components 𝐶 of 𝑆𝑖 ∩ 𝑆 𝑗 , such that

(𝜕1

𝐶 (𝛾𝑖), 𝜕1

𝐶 (𝛾 𝑗 )) = (0, 0), and 𝛾𝑖 |𝐶 and 𝛾 𝑗 |𝐶 are not both trivial in 𝐻1(𝑘 (𝐶), Z/3).

Then 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑘, Z/3) contains the subquotient 𝐻′/⟨1, · · · , 1⟩.

Proof. First, note that under these assumptions, 𝑌 is necessarily integral (see Corollary 4.2.6),

hence we can talk about its function field 𝑘 (𝑌 ). We have the following commutative diagram:

0

Z/3

0

𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑌 , Z/3)

𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝐾, Z/3)

⊕𝜕2
𝑇

(cid:202)

𝐻1(𝑘 (𝑇), Z/3)

𝜎

Br𝑛𝑟 (𝐾) [3] = 0

𝐻2(𝐾, Z/3)

⊕𝜕2
𝑆

𝑇 ∈𝑌 (1)
𝐵

(cid:202)

𝑆 ∈ 𝐵(1)

𝜏

𝐻1 (𝑘 (𝑆), Z/3)

⊕𝜕1
𝐶

(cid:202)

Z/3

𝐶 ∈ 𝐵(2)

Γ

0

⟨𝛼⟩

0

17

We make some observations related to this diagram:

1. By definition, 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑌 , Z/3) denotes all those classes in 𝐻2(𝑘 (𝑌 ), Z/3) which are un-

ramified with respect to divisorial valuations corresponding to prime divisors on 𝑌 , Since

the singular locus of 𝑌 has codimension ≥ 2, we can also characterize 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑌 , Z/3) as

all those classes in 𝐻2(𝑘 (𝑌 ), Z/3) that are unramified with respect to divisorial valuations

which have centers on Y which are not contained in 𝑌𝑠𝑖𝑛𝑔 [ABvBP20, Cor. 2.2].

2. 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝐾, Z/3) denotes those classes in 𝐻2(𝑘 (𝑌 ), Z/3) which are killed by residue maps

associated to divisorial valuations that are trivial on 𝐾, hence correspond to prime divisors of
𝑌 dominating the base 𝐵. We use 𝑌 (1)
𝐵

to denote all prime divisors in 𝑌 that do not dominate

the base 𝐵. Then the upper row is exact by definition.

3. The second row is obtained from Bloch-Ogus complex [BO74], which is exact under the

assumptions

Br(𝐵) [3] = 0, and 𝐻3

ét(𝐵, Z/3) = 0.

4. The left vertical row is exact by Lemma 3.2.4, because we have

𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝐾, Z/3) (cid:27) 𝐻2

𝑛𝑟 (𝐾 (𝑆0)/𝐾, Z/3) (cid:27) 𝐵𝑟 (𝑆0) [3],

where 𝑆0 is the Brauer-Severi surface (over 𝐾) corresponding to the generic fiber 𝛼. Hence

𝑆0 is smooth and we have the last isomorphism in the above statement.

5. In the right vertical row, the map 𝜏 is induced by the field extensions 𝑘 (𝑆) ⊂ 𝑘 (𝑇), coincides

with the induced map

𝑘 (𝑆)×/𝑘 (𝑆)×3 → 𝑘 (𝑇)×/𝑘 (𝑇)×3.

If 𝑆 is not contained in the discriminant locus, the generic fiber of 𝑇 → 𝑆 is geometrically

integral. Then 𝑘 (𝑆) is algebraically closed in 𝑘 (𝑇), and thus the induced map above is

injective.

If 𝑆 = 𝑆𝑖 is a component of the discriminant locus, then after taking the base

change to the cubic extension 𝐹/𝑘 (𝑆𝑖) defined by the residue class 𝛾𝑖 ∈ 𝐻1(𝑘 (𝑆𝑖), Z/3), the

18

generic fiber of 𝑇𝑖 → 𝑆𝑖 is a union of three Hirzebruch surfaces F1, meeting transversally

so that any pair of them meet along a fiber of one and a (−1)-curve of the other. (This is

correct because 𝑇𝑖 is the unique irreducible componenet dominate 𝑆𝑖 in the preimage of 𝑆𝑖

under 𝜋, with the third assumption in Definition 3.2.1). In this case, the low degree long

exact sequence from the Hochschild-Serre spectral sequence

𝐻 𝑝 (Gal(𝐹/𝑘 (𝑆𝑖)), 𝐻𝑞 (Spec(𝐹), Z/3)) ⇒ 𝐻 𝑝+𝑞 (Spec(𝑘 (𝑆𝑖)), Z/3)

implies the kernel of the natural map 𝐻1(𝑘 (𝑆𝑖), Z/3) → 𝐻1(𝐹, Z/3) is generated by 𝛾𝑖. By

the last assumption in Definition 3.2.1, we know ker(𝜏) = Γ.

Then we can prove that 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑌 , Z/3) lies in the image of 𝜎. In fact, let 𝜉 ∈ 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑌 , Z/3)

and denote by 𝜉 again its the image in 𝐻2

𝑇 . If 𝜉 is not in
the image of 𝜎, it lifts to a class 𝜉 ′ ∈ 𝐻2(𝐾, Z/9) by Lemma 3.2.4. We have the following exact

𝑛𝑟 (𝑘 (𝑌 )/𝑌 , Z/3). Then 𝜉 is killed by ⊕𝜕2

sequence which is similar to the second row in above diagram with coefficients Z/9:

0 → 𝐻2(𝐾, Z/9)

⊕𝜕2
𝑆−−−→

(cid:202)

𝑆∈𝐵 (1)

𝐻1(𝑘 (𝑆), Z/9)

⊕𝜕1
𝐶−−−→

(cid:202)

Z/9

𝐶∈𝐵 (2)

Hence at least one residue 𝜕2

𝑆 (𝜉 ′) must have order 9 (since ⊕𝜕2

𝑆 is injective both for 3-torsion and

9-torsion cases ). On the other hand,

ker(

(cid:202)

𝑆∈𝐵 (1)

𝐻1(𝑘 (𝑆), Z/9) →

(cid:202)

𝑇 ∈𝑌 (1)
𝐵

𝐻1(𝑘 (𝑇), Z/9)) (cid:27) Γ

This is correct because (again we use 𝐹 to denote a separable closure of 𝑘 (𝑆𝑖)) in the long exact

sequence associate to

𝐻 𝑝 (Gal(𝐹/𝑘 (𝑆𝑖)), 𝐻𝑞 (Spec(𝐹), Z/9)) ⇒ 𝐻 𝑝+𝑞 (Spec(𝑘 (𝑆𝑖)), Z/9)

We have

0 → 𝐻1(𝐺𝑎𝑙 (𝐹/𝑘 (𝑆𝑖)), Z/9) → 𝐻1(Spec(𝑘 (𝑆𝑖)), Z/9) → 𝐻1(Spec(𝐹), Z/9)

19

As we can calculate étale cohomology of spectrum of a field using Galois cohomology, we have

the kernel:

𝐻1(𝐺𝑎𝑙 (𝐹/𝑘 (𝑆𝑖)), Z/9) (cid:27) 𝐻𝑜𝑚𝑐𝑜𝑛𝑡 (Z/3, Z/9) (cid:27) Z/3

While the kernel for those 𝑆 doesn’t belongs to the discriminant locus is clearly zero by the same

argument in 3-torsion case.

Now we notice that Γ has no elements of order 9, this means 𝜕2

𝑆 (𝜉 ′) can’t be mapped to 0 in

(cid:201)

𝑇 ∈𝑌 (1)
𝐵

𝐻1(𝑘 (𝑇), Z/3), hence a contradiction.

The above diagram chasing in fact gives us

𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑌 , Z/3) (cid:27) Γ ∩ ker(⊕𝜕1

𝐶)/⟨𝛼⟩ (cid:27) 𝐻/⟨𝛼⟩ .

Next we determine classes in 𝐻 that are in 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑘, Z/3). In particular, we show that the

subgroup 𝐻′ defined earlier is contained in 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑘, Z/3). We do this by checking whether the

classes in 𝐻′ are unramified with respect to all divisorial valuations 𝜇 of 𝑘 (𝑌 ) (and not just those

that come from prime divisors on 𝑌 ). Consider a class 𝛽 ∈ 𝐻, viewed as an element in 𝐻2(𝐾, Z/3).

Denote by 𝛽′ the image of 𝛽 in 𝐻2(𝑘 (𝑌 ), Z/3). We aim to show that 𝛽′ is unramified on 𝑌 if 𝛽 is

in 𝐻′. Using the definition of 𝐻, it is sufficient to check this for valuations whose centers on 𝐵 has

codimension at least 1. In the following, we use 𝒪 to denote the local ring of 𝜇 in 𝐵.

Case 1: The center of 𝜇 on 𝐵 is not contained in the discriminant locus: In this case, for any surface

𝑆 passing through the center of 𝜇, we have

𝜕2
𝑆 (𝛽) = 0.

Then [ABvBP20, Proposition 2.1] tells us 𝛽 is in the image of 𝐻2

ét(𝒪, Z/3). Hence 𝛽′ = 𝜎(𝛽)

is also unramified with respect to 𝜇 in this case.

Case 2: The center of 𝜇 on 𝐵 is contained in the discriminant locus, but not in the intersection of two

or more components: Now the center is contained in 𝑆𝑖 for a unique 𝑖. Recall that the 𝑖𝑡ℎ

component 𝑥𝑖 of ⊕𝜕2

𝑆 (𝛽) is 0, 1 or 2. If 𝑥𝑖 = 0, by an argument same as Case 1, 𝛽 is in the

20

image of 𝐻2

ét(𝒪, Z/3). Similarly, if 𝑥𝑖 = 1, 𝛽 − 𝛼 is in the image of 𝐻2

ét(𝒪, Z/3). Finally, if

𝑥𝑖 = 2, 𝛽 − 2𝛼 is in the image of 𝐻2

ét(𝒪, Z/3). Notice that

𝛽′ = 𝜎(𝛽) = 𝜎(𝛽 − 𝛼) = 𝜎(𝛽 − 2𝛼).

So in all three conditions, we have 𝛽′ is unramified with respect to 𝜇 in this case.

Case 3: The center 𝜇 on 𝐵 is a curve 𝐶 that is an irreducible component of 𝑆𝑖 ∩ 𝑆 𝑗 : In this case, we

again check the possible values of 𝑥𝑖 and 𝑥 𝑗 in ⊕𝜕2

𝑆 (𝛽). We have the following cases:

Case 3(a): If 𝑥𝑖 = 𝑥 𝑗 , then the argument in Case 2 above gives us that at least one of 𝛽, 𝛽 − 𝛼 or

𝛽 − 2𝛼 lies in the image of 𝐻2

ét(𝒪, Z/3). So we are done in this situation.

Case 3(b): (𝑥𝑖, 𝑥 𝑗 ) = (0, 1) or (1, 0): By symmetry, we can assume (𝑥𝑖, 𝑥 𝑗 ) = (1, 0). Notice that

(cid:16)

(cid:12)
(cid:12)
(cid:12)

3

𝐶 (𝛾𝑖) + 𝜕1
𝜕1

𝐶 (𝛾 𝑗 )

(cid:17)

by the exactness of the second row in the diagram. Then we must have

𝐶 (𝛾𝑖) = 𝜕1
𝜕1

𝐶 (𝛾 𝑗 ) = 0

This means that a rational function representing the class

𝛾𝑖 ∈ 𝐻1(𝑘 (𝑆𝑖), Z/3) = 𝑘 (𝑆𝑖)×/𝑘 (𝑆𝑖)×3

has a zero or a pole of order divisible by 3 along 𝐶. Without loss of generality, we

may assume that the function associated with 𝛾𝑖 is contained in the local ring 𝒪𝑆𝑖,𝐶 of

𝐶 in 𝑆𝑖.We call this function 𝑓𝛾𝑖 . Let 𝑡 be a local parameter for 𝐶 in 𝒪𝑆𝑖,𝐶. Such a

local parameter exists as 𝐶 is a Cartier divisor on 𝑆𝑖, which in turn follows since 𝑆𝑖 is
𝑓𝛾𝑖

assumed to be factorial along 𝐶. Then

𝑡 𝜇𝐶 ( 𝑓𝛾𝑖 ) is a unit, and hence any preimage in 𝒪 is
also a unit (See Remark 3.3.1 below). Call such a preimage 𝑢𝛾𝑖 , which may be viewed

as a rational function in 𝐾. Assume 𝜋𝑆𝑖 is a local parameter of 𝑆𝑖 in 𝒪. Consider the

symbol algebra (𝑢𝛾𝑖 , 𝜋𝑆𝑖 ) ∈ 𝐻2(𝐾, Z/3). Let 𝑆 be a surface containing 𝐶. By Lemma

21

2.3.1, we have

𝜕2

𝑆 (𝑢𝛾𝑖 , 𝜋𝑆𝑖 ) = (−1) 𝜇𝑆 (𝑢𝛾𝑖 )𝜇𝑆 (𝜋𝑆𝑖 ) ¯𝑢𝜇𝑆 (𝜋𝑆𝑖 )
𝛾𝑖
𝜋𝜇𝑆 (𝑢𝛾𝑖 )
𝑆𝑖

=

¯𝑢𝛾𝑖

if 𝑆 = 𝑆𝑖

0

if 𝑆 ≠ 𝑆𝑖





On the other hand, ¯𝑢𝛾𝑖 = 𝛾𝑖 by construction, so we have

𝜕2
𝑆𝑖 (𝑢𝛾𝑖 , 𝜋𝑆𝑖 ) = 𝛾𝑖 = 𝜕2

𝑆𝑖 (𝛽)

𝜕2
𝑆 𝑗 (𝑢𝛾𝑖 , 𝜋𝑆𝑖 ) = 0 = 𝜕2

𝑆 𝑗 (𝛽)

Also 𝜕2

𝑆 (𝛽) = 0 if 𝑆 is a surface passing through 𝐶 other than 𝑆𝑖 and 𝑆 𝑗 .(In fact, by
our assumption, such an 𝑆 is not in the discriminant locus and so this agrees with this

conclusion.) Hence [ABvBP20, Proposition 2.1] tells us 𝛽 − (𝑢𝛾𝑖 , 𝜋𝑆𝑖 ) is in the image

of 𝐻2

ét(𝒪, Z/3). Hence

It then suffices to show that

𝜕2
𝜇 (𝜎(𝛽 − (𝑢𝛾𝑖 , 𝜋𝑆𝑖 ))) = 0

𝜇 (𝜎((𝑢𝛾𝑖 , 𝜋𝑆𝑖 ))) = ± ¯𝑢𝜇(𝜋𝑆𝑖 )
𝜕2

𝛾𝑖

= 0 ∈ 𝐻1(𝑘 (𝜇), Z/3)

By assumption, ¯𝑢𝛾𝑖 |𝐶 is trivial, hence so is ¯𝑢𝜇(𝜋𝑆𝑖 )

𝛾𝑖

as the center of 𝜇 is 𝐶.

Case 3(c): (𝑥𝑖, 𝑥 𝑗 ) = (0, 2) or (2, 0): By symmetry, we can assume (𝑥𝑖, 𝑥 𝑗 ) = (2, 0). Now the

proof is essentially same as Case 3(b), which shows that

𝜇 (𝜎(𝛽 − 2(𝑢𝛾𝑖 , 𝜋𝑆𝑖 ))) = 0.
𝜕2

It follows that 𝜕2

𝜇 (𝜎(𝛽)) = 0. and so 𝛽′ is unramified along 𝜇.

Case 3(d): (𝑥𝑖, 𝑥 𝑗 ) = (1, 2) or (2, 1): Assume (𝑥𝑖, 𝑥 𝑗 ) = (2, 1). Then applying Case 3(𝑏) to the

class 𝛽 − 𝛼, we see that this case is also proved.

Case 4: The center of 𝜇 on 𝐵 is a point 𝑃 ∈ 𝐶, here 𝐶 is as in case 3, and 𝑆𝑖, 𝑆 𝑗 are the only surfaces

among the 𝑆1, · · · , 𝑆𝑛 that pass through 𝑃. As we have seen in the discussion of Case 3, we

can reduce to the case when (𝑥𝑖, 𝑥 𝑗 ) = (1, 0). Hence we again have 𝜕1

𝐶 (𝛾𝑖) = 𝜕1

𝐶 (𝛾 𝑗 ) = 0.

22

In fact, this is true for any curve 𝐶′ that contains 𝑃 and is contained in 𝑆𝑖 ∪ 𝑆 𝑗 . Choose a

function 𝑓𝛾𝑖 ∈ 𝑘 (𝑆𝑖) representing the class 𝛾𝑖. Then let 𝐶1, · · · , 𝐶𝑁 be all irreducible curves

through 𝑃 that are either a zero or a pole for the function 𝑓𝛾𝑖 . Pick local equations 𝑡ℓ of 𝐶ℓ in

𝒪𝑆𝑖,𝑃, and consider the following rational function on 𝑆𝑖:

𝑓𝛾𝑖
· · · 𝑡

.

(cid:17)

𝜇𝐶𝑁 ( 𝑓𝛾𝑖 )
𝑁

( 𝑓𝛾𝑖 )

(cid:16)

𝑡

𝜇𝐶
1
1

Since 𝑆𝑖 is assumed to be factorial, in particular, normal at 𝑃, the above rational function is a

unit locally around 𝑃. Hence it can be lifted to a unit in 𝒪. Then we can repeat the rest of the

proof as in Case 3(b). (Notice that every element in 𝑘 (𝑃) is a cube since 𝑘 is algebraically

closed, so the last step of Case 3(b) is automatically true.)

Case 5: The center of 𝜇 on 𝐵 is a point 𝑃 that lies on exactly three distinct surfaces 𝑆𝑖, 𝑆 𝑗 , 𝑆𝑙: We

consider the possible values of (𝑥𝑖, 𝑥 𝑗 , 𝑥𝑙). If 𝑥𝑖 = 𝑥 𝑗 = 𝑥𝑙, then one of 𝛽, 𝛽 − 𝛼 or 𝛽 − 2𝛼

is unramified. By symmetry and up to subtraction by 𝛼 or 2𝛼, the only remaining cases are

(1, 0, 0), (1, 1, 0), and (2, 1, 0). Notice that (2, 1, 0) = (1, 1, 0) + (1, 0, 0), and that the case

(1, 1, 0) is equivalent to the case (2, 0, 0). Hence, we only need to consider the case (1, 0, 0),

which is same as Case 4. Now the rest of the proof is same as in Case 4.

□

Remark 3.3.1. In Case 3(b) in Theorem 3.3.1, we claimed that if ¯𝑥 ∈ 𝒪𝑆𝑖,𝐶 is a unit, then any

preimage 𝑥 in 𝒪 = 𝒪𝐵,𝐶 is also a unit. In fact, we have

𝒪𝑆𝑖,𝐶 (cid:27) 𝒪/(𝜋𝑆𝑖 ).

As ¯𝑥 is a unit in 𝒪𝑆𝑖,𝐶, there exist a ¯𝑦 ∈ 𝒪𝑆𝑖,𝐶 such that ¯𝑥 ¯𝑦 = 1 ∈ 𝒪𝑆𝑖,𝐶. Hence there exist 𝑡 ∈ 𝒪,

such that

𝑥𝑦 = 1 + 𝜋𝑆𝑖 𝑡 ∈ 𝒪

Notice that 𝜋𝑆𝑖 𝑡 is contained in the maximal ideal of 𝒪, so 1 + 𝜋𝑆𝑖 𝑡 is a unit in 𝒪. Hence any

preimage 𝑥 is also a unit in 𝒪.

23

We prove an immediate corollary in which we weaken the hypothesis about factoriality when

𝑛 = 2. In this case, the discriminant locus has exactly two irreducible components. We prove that

it is sufficient to have only one of them factorial at their intersection to make the unramified Brauer

group nontrivial:

Corollary 3.3.2. Assume 𝑛 = 2. We continue with the same hypothesis as in the theorem except

the following change: we replace the requirement (3) by the following:

(3’) 𝑆1 is factorial at every point of 𝑆1 ∩ 𝑆2 .

Then 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑘, Z/3) is nontrivial and hence 𝑌 is not stably rational.

Proof. In this case, the Brauer class 𝛽 in 𝐻′ whose representative is (1, 0) can be lifted to a

nontrivial unramified Brauer class in 𝐻2(𝑘 (𝑌 )/𝑘, Z/3)

□

24

CHAPTER 4

FLATNESS AND EXAMPLES

4.1 A distinguished example

In this section, we will construct a Brauer-Severi surface bundle over P3 that is stably non-

rational. We use Corollary 3.3.2 for this purpose.

Example 4.1.1. Consider the following two surfaces in P3

C = Proj C[𝑥0, 𝑥1, 𝑥2, 𝑥3]:

𝑆1 : {𝑥9

0 + (𝑥3

1 − 𝑥3

2) (𝑥3

2 − 𝑥3

3) (𝑥3

3 − 𝑥3

1) = 0}

(cid:16)

𝑆2 : {

0 + (𝑥3
𝑥9

1 − 𝑥3

2)(𝑥3

2 − 𝑥3

3) (𝑥3

3 − 𝑥3
1)

(cid:17) (cid:16)

0 − 𝑥3
𝑥9

1

𝑥3
2

𝑥3
3

(cid:17)

+ 𝑥6
1

𝑥6
2

𝑥6
3 = 0}

In the following, we use 𝐹𝑆1
checking that both 𝑆1 and 𝑆2 are irreducible and reduced:

, 𝐹𝑆2 to denote the equation defines 𝑆1, 𝑆2 separately. We start by

• 𝑆1 is irreducible and reduced. This follows directly from the fact that the singular locus of

𝑆1 has dimension 0. In fact, 𝑆1 only singular at 12 isolated points:

[0 : 1 : 0 : 0] , [0 : 0 : 1 : 0] , [0 : 0 : 0 : 1]

[0 : 𝜔 : 1 : 1] , [0 : 1 : 𝜔 : 1] , [0 : 1 : 1 : 𝜔]

[0 : 𝜔2 : 1 : 1], [0 : 1 : 𝜔2 : 1], [0 : 1 : 1 : 𝜔2]

[0 : 𝜔2 : 𝜔 : 1] , [0 : 𝜔 : 𝜔2 : 1] , [0 : 1 : 1 : 1]

Here 𝜔 is a primitive 3𝑟𝑑 roots of unity. If 𝑆1 is not reduced, then the singular locus would
have dimension 2. If 𝑆1 is not irreducible, the singular locus would have dimension at least

1 by Bèzout theorem.

• 𝑆2 is irreducible and reduced. We may rewrite the equation defining 𝑆2 as:

0 + 𝑃(𝑥1, 𝑥2, 𝑥3)𝑥9
𝑥18

0 − 𝑥3

1

𝑥3
2

𝑥3
3

𝑃(𝑥1, 𝑥2, 𝑥3)

where 𝑃(𝑥1, 𝑥2, 𝑥3) = (𝑥3
1

𝑥3
) (𝑥3
3. We may consider the above
3
polynomial as an element in C[𝑥0, 𝑥1, 𝑥2, 𝑥3] = C[𝑥1, 𝑥2, 𝑥3] [𝑥0], which is a UFD. Hence to

) − 𝑥3
1

− 𝑥3
1

− 𝑥3
3

− 𝑥3
2

)(𝑥3
2

𝑥3
2

25

check it is irreducible, it is sufficient to use Eisenstein’s criterion: We need to find a prime

ideal 𝔭 in C[𝑥1, 𝑥2, 𝑥3] [𝑥0], such that

𝑃(𝑥1, 𝑥2, 𝑥3) ∈ 𝔭,

𝑥3
1

𝑥3
2

𝑥3
3

𝑃(𝑥1, 𝑥2, 𝑥3) ∈ 𝔭 and

𝑥3
1

𝑥3
2

𝑥3
3

𝑃(𝑥1, 𝑥2, 𝑥3) ∉ 𝔭2.

It is evident that an appropriate prime ideal exists if 𝑃(𝑥1, 𝑥2, 𝑥3) has an irreducible factor

with multiplicity 1 and is coprime to 𝑥1𝑥2𝑥3. In fact, any irreducible factor of 𝑃(𝑥1, 𝑥2, 𝑥3)

is inherently coprime to 𝑥1𝑥2𝑥3. Therefore it suffices to provide a single regular point of

𝑃(𝑥1, 𝑥2, 𝑥3) to show the existence of such an irreducible factor. Finally, we directly check

that (1, 1, 0) is a regular point of 𝑃(𝑥1, 𝑥2, 𝑥3). Hence 𝑆2 is irreducible.

Now as we have already shown 𝑆2 is irreducible, it is sufficient to find a smooth point in 𝑆2 to
show it is reduced. Indeed, one can easily check that [(−56) 1

9 : 1 : 2 : 0] is indeed a smooth

point of 𝑆2.

We choose rational triple covers of 𝑆1 and 𝑆2 defined by:

𝛾1 =

𝑥3
2

− 𝑥3
3
𝑥3
0

∈ 𝐻1(C(𝑆1), Z/3) (cid:27) C(𝑆1)×/C(𝑆1)×3

𝛾2 =

𝑥3
2

𝑥3
3

𝑥9
0

− 𝑥3
1
𝑥9
0

∈ 𝐻1(C(𝑆2), Z/3) (cid:27) C(𝑆2)×/C(𝑆2)×3

We claim the triple covers 𝛾1, 𝛾2 are not trivial: In fact, by Lemma 2.3.1, the residue of 𝛾1 of a

valuation centered at the point [0 : 𝜔 : 1 : 1] is 1 ∈ Z/3. Hence 𝛾1 is not trivial. To show 𝛾2 is not

trivial is equivalent to show 𝐹𝑆1 is not a cubic in the function field of 𝑆2. And it’s true because the
of a valuation centered at the point [0 : 0 : 1 : 1] is 1 ∈ Z/3. Hence 𝛾1, 𝛾2 are not

residue of

𝐹𝑆
1
𝑥9
0

trivial.

Consider the corresponding Bloch-Ogus exact sequence:

0

𝐵𝑟 (C(P3

C)) [3]

⊕𝜕2
𝑆

(cid:202)

𝐻1(𝑘 (𝑆), Z/3)

⊕𝜕1
𝐶

(cid:202)

𝐻0(𝑘 (𝐶), Z/3)

𝑆∈(P3

C) (1)

𝐶∈(P3

C) (2)

26

We have ⊕𝜕1

𝐶 (𝛾1) = ⊕𝜕1

𝐶 (𝛾2) = 0. In fact, it is easy to check that for any curve 𝐶 such that 𝛾1(or

𝛾2) has a zero or pole along 𝐶, the order is divided by 3. Hence

(1, · · · , 1, 𝛾1, 1, · · · , 1, 𝛾2, 1, · · · ) ∈

(cid:202)

𝑆∈(P3

C) (1)

𝐻1(𝑘 (𝑆), Z/3)

can be lifted to a Brauer class [𝒜] = [(

𝐹𝑆
2

−𝑥3
3

(𝑥3
2
𝑥21
0

)

, 𝐹𝑆
1
𝑥9
0

)𝜔] ∈ Br(C(P3

C)) [3]. This can be directly

checked by Lemma 2.3.1.

By Theorem 4.2.1, the cyclic algebra 𝒜 gives out a Brauer-Severi surface bundle 𝑌 → P3
C.

This Brauer-Severi surface bundle has a good discriminant locus. We prove this by checking the

conditions in Definition 3.2.1. Here is the list of corresponding arguments:

1. We already proved that 𝑆1 and 𝑆2 are reduced.

2. The behavior of a general fiber over 𝑆1 and 𝑆2 is given by [Mae97, Thm. 2.1].

3. The induced triple cover over 𝑆1 and 𝑆2 are irreducible because 𝛾1, 𝛾2 are not trivial.

4. To show the last requirement in Definition 3.2.1 is true, we have the following commutative

diagram:

𝑎

𝐹×/𝐹×3
𝑏
𝑘 (𝑆𝑖)×/𝑘 (𝑆𝑖)×3

𝐹 (𝑢, 𝑣)×/𝐹 (𝑢, 𝑣)×3
𝑑
𝑘 (𝑇𝑖)×/𝑘 (𝑇𝑖)×3

𝜏𝑖

Where 𝑇𝑖 is defined right after the large diagram in Theorem 3.3.1. For 𝑆1, 𝑏 is induced by
)

the cubic extension defined by 𝛾1, 𝑑 is induced by the cubic extension defined by

−𝑥3
3

𝐹𝑆
2

,

(𝑥3
2
𝑥21
0

which is equal to the cubic class defined by 𝛾1 ([Art82a, Thm. 2.1]. ) Note that 𝑎 is injective,

and ker(𝑏) =< 𝛾1 >. On the other hand, a diagram chasing as in part (5) in proof of Theorem

3.3.1 shows that ker(𝜏1) contains < 𝛾1 > . This forces 𝑑 to be injective and ker(𝜏1) =< 𝛾1 >.

Same argument works for 𝑆2.

On the other hand, we list all irreducible components of 𝑆1 ∩ 𝑆2:

𝐷 = 6𝐷1 + 6𝐷2 + 6𝐷3

27

Where

𝐷1 = {𝑥1 = 0, 𝑥9

0 − 𝑥6

2

𝐷2 = {𝑥2 = 0, 𝑥9

0 − 𝑥6

3

𝐷3 = {𝑥3 = 0, 𝑥9

0 − 𝑥6

1

3 + 𝑥3
𝑥3

2

𝑥6
3 = 0}

𝑥3
1 + 𝑥3

3

𝑥6
1 = 0}

2 + 𝑥3
𝑥3

1

𝑥6
2 = 0}

One can easily check they are indeed irreducible using Eisenstein’s criterion by viewing those

polynomials as elements in C[𝑥1, 𝑥2, 𝑥3] [𝑥0]. Notice that 𝐷1 passes through only two singular

points of 𝑆1: [0 : 0 : 1 : 0] and [0 : 0 : 0 : 1]. It is straightforward to check 𝐷1 is indeed a Cartier

divisor of 𝑆1, even along these two singular points:

Lemma 4.1.2. 𝐷𝑖 are Cartier divisors of 𝑆1.

Proof. By symmetry, it is sufficient to check the behavior of 𝐷1 at singular points of 𝑆1. Notice

that 𝐷1 only passes through two singular points of 𝑆1: [0 : 0 : 1 : 0] and [0 : 0 : 0 : 1]. Let

𝑃 = [0 : 0 : 0 : 1], then we have the local ring:

𝒪𝑆1,𝑃 = (C[𝑥0, 𝑥1, 𝑥2]/(𝑥9

0 + (𝑥3

1 − 𝑥3

2) (𝑥3

2 − 1) (1 − 𝑥3

1)))(𝑥0,𝑥1,𝑥2)

By expanding the equation defining 𝑆1, we have

0 − 𝑥6
𝑥9

2 + 𝑥3

2 = 𝑥3

1 (𝑥3

1

2 − 𝑥6
𝑥3

2 − 𝑥3

1 + 1) ∈ 𝒪𝑆1,𝑃

Notice 𝑥3
1

𝑥3
2

− 𝑥6
2

− 𝑥3
1

+ 1 is a unit in 𝒪𝑆1,𝑃, hence the ideal defining 𝐷1, which is (𝑥1, 𝑥9
0

− 𝑥6
2

+ 𝑥3
2

),

is generated by one element 𝑥1. Similarly one can do the calculation for the point [0 : 0 : 1 : 0].

As a result, 𝐷1 is a Cartier divisor of 𝑆1.

□

Finally, we need to check both 𝛾1|𝐷𝑖 and 𝛾2|𝐷𝑖 are trivial for 𝑖 ∈ {1, 2, 3}. These are directly

following from the choices of 𝛾1 and 𝛾2. Hence in this example, using notations in Theorem 3.3.1,

we have 𝐻′ = 𝐻 = Γ = Z/3 × Z/3. By Corollary 3.3.2, the unramified Brauer group of 𝑌 contains

a subgroup Z/3, hence 𝑌 is not stably rational.

28

4.2 Flatness

In this section, we check the cyclic algebra

𝒜 = (

− 𝑥3
3

𝐹𝑆2 (𝑥3
2
𝑥21
0

)

,

𝐹𝑆1
𝑥9
0

)𝜔

indeed gives us a Brauer-Severi surface bundle over P3

C as in Definition 1.0.6. We keep the notation

in Example 4.1.1 through out this section. The definition of a general Brauer-Severi scheme is

given by Van den Bergh in [VdB88].

In [See99], Seelinger gave an alternating description of

Brauer-Severi scheme which is easier to use in our case. See also Section 1 in [Mae97] for the

discussion of the following definitions:

Definition 4.2.1. Let Λ be a sheaf of 𝒪P3

C

algebra that is torsion free and coherent as an 𝒪P3

C

module.

We say Λ is an 𝒪P3

C

-order in 𝒜 if Λ contains 𝒪P3

C

and

Λ ⊗𝒪P3

C

C(P3

C) (cid:27) 𝒜

Definition 4.2.2. For each point 𝑝 ∈ P3

C, let 𝒪P3

a finitely generated 𝒪P3

C,𝑝 algebra Λ𝑝 is an 𝒪P3

C,𝑝 denote the regular local ring of P3
C,𝑝-order in 𝒜, if Λ𝑝 is torsion free and

C at 𝑝. We say

Λ𝑝 ⊗𝒪P3

C

, 𝑝

C(P3

C) (cid:27) 𝒜

Remark 4.2.3. In this paper, we always assume an order is locally free.

Recall that (3.4) of [Mae97] describes an 𝒪P3

C

-order which we again denote by Λ in the following,

we denote its localization at a point 𝑝 by Λ𝑝.

Definition 4.2.4. Let 𝑉Λ (respectively, 𝑉Λ 𝑝 ) be the functor from the category of P3

C-schemes

(respectively, Spec(𝒪P3

C,𝑝)-schemes) to the category of sets:

𝑉Λ(𝑆) = {[𝑧] ∈ 𝐺𝑛 [(Λ ⊗𝒪P3

C

𝑆)∨] | 𝑧 · 𝑢 = 𝑁𝑆 (𝑢)𝑧 , ∀𝑢 ∈ (Λ ⊗𝒪P3

C

𝑆)∗}

𝑉Λ 𝑝 (𝑆) = {[𝑧] ∈ 𝐺𝑛 [(Λ𝑝 ⊗𝒪P3

C

, 𝑝

𝑆)∨] | 𝑧 · 𝑢 = 𝑁𝑆 (𝑢)𝑧 , ∀𝑢 ∈ (Λ ⊗𝒪P3

C

𝑆)∗}

, 𝑝

29

where ∨ denotes the dual sheaf, ∗ denotes the unit group , 𝑁𝑆 is the reduced norm and 𝐺𝑛 denotes

the functor of Grassmannian of 𝑛-quotients([VdB88, Def.1]). These functors are represented by

schemes as these are closed subschemes of the Grassmannian, which we call the Brauer-Severi

scheme (associated to Λ, Λ𝑝) and again denote them by 𝑉Λ, 𝑉Λ 𝑝 .

Theorem 4.2.1. 𝑌 = 𝑉Λ is a Brauer-Severi surface bundle over P3
C.

Proof. According to Definition 4.2.4, for every closed point 𝑝 in P3

C, we have the following

commutative diagram of schemes:

𝑉Λ ×P3

C

C,𝑝) (cid:27) 𝑉Λ 𝑝

Spec(𝒪P3
𝜋 𝑝
C,𝑝)

Spec(𝒪P3

𝑉Λ
𝜋

P3
C

We first show 𝜋 is a flat morphism. In order to do so, it suffices to show 𝜋 𝑝 is flat for all closed

points 𝑝 ∈ P3

C. Indeed, if this is done, the flat locus of 𝜋 would be an open subset of P3

C containing

all closed points, hence is equal to P3

the fact that Spec(𝒪P3

C. Furthermore, by the "Miracle flatness" theorem [sta23] and
C,𝑝) is regular, it suffices to show each 𝑉Λ 𝑝 is Cohen-Macaulay and each fiber
of 𝜋 𝑝 has the same dimension. We do this by a case-by-case argument for all closed points in P3
C:

Case 1: 𝑝 ∉ 𝑆1 ∪ 𝑆2. It is well know that Λ is an Azumaya algebra outside of discriminant locus

[Art82a]. All fibers of 𝜋 𝑝 are smooth Brauer-Severi surfaces and furthermore 𝑉Λ 𝑝 is regular,

hence Cohen-Macaulay. By the "Miracle flatness" theorem, 𝜋 𝑝 is flat in this case.

Case 2: 𝑝 ∈ 𝑆1 ∪ 𝑆2 and 𝑝 ∉ 𝑆1 ∩ 𝑆2 and 𝑝 ∉ 𝑆1 ∩ {𝑥3
2

− 𝑥3

[Art82a], we may write Λ𝑝 as the symbol algebra ( 𝑓𝑝, 𝑔𝑝)𝜔. That is, Λ𝑝 over Spec(𝒪P3

3 = 0} . Following ideas from Artin
C,𝑝) is

generated by 𝑥, 𝑦 subject to the relations

𝑥3 = 𝑓𝑝, 𝑦3 = 𝑔𝑝, 𝑥𝑦 = 𝜔𝑦𝑥.

Since 𝑝 ∉ 𝑆1 ∩ {𝑥3
2

then

−𝑥3

3 = 0}, it follows that 𝑓𝑝 is a unit in 𝒪P3

C,𝑝. Let 𝑅𝑝 = 𝒪P3

C,𝑝 [𝑇]/(𝑇 3 − 𝑓𝑝),

Spec(𝑅𝑝)

𝜏
−→ Spec(𝒪P3

C,𝑝)

30

is an étale neighborhood of Spec(𝒪P3

C,𝑝) with 𝜏 faithfully flat as it surjects on the underlying
topological space. By faithfully flat descent, it suffices to show 𝑉Λ 𝑝 ⊗ 𝑅𝑝 is flat over Spec(𝑅𝑝).

In [Art82a], Artin noticed 𝑉Λ 𝑝 ⊗ 𝑅𝑝 can be viewed as a subalgebra of the 3 by 3 matrices

algebra over 𝑅𝑝 by setting

𝑥 =

0 𝑇𝜔

0

0


𝑇










0

And 𝑉Λ 𝑝 ⊗ 𝑅𝑝 can be embedded into P2
𝑅 𝑝

cyclic permutations in indices:

, 𝑦 =

0

0











× P2
𝑅 𝑝

𝑇𝜔2

0

0

0 1

1 0











× P2
𝑅 𝑝 by the following 9 equations with a

𝑔𝑝 0 0












𝑔𝑝𝜉11𝜉22 = 𝜉12𝜉21

𝑔𝑝𝜉11𝜉23 = 𝜉13𝜉21

𝑔𝑝𝜉11𝜉32 = 𝑔𝑝𝜉12𝜉31

𝑔𝑝𝜉11𝜉33 = 𝜉13𝜉31

𝜉12𝜉23 = 𝜉13𝜉22

𝑔𝑝𝜉12𝜉33 = 𝜉13𝜉32

𝑔𝑝𝜉21𝜉32 = 𝑔2

𝑝𝜉22𝜉31

𝑔𝑝𝜉21𝜉33 = 𝑔2

𝑝𝜉23𝜉31

𝑔𝑝𝜉22𝜉33 = 𝜉23𝜉32

Here we use [𝜉11 : 𝜉12 : 𝜉13], [𝜉21 : 𝜉22 : 𝜉23], [𝜉31 : 𝜉32 : 𝜉33] to denote the coordinates in

× P2

× P2
𝑅 𝑝

P2
𝑅 𝑝 . Note that even though Artin’s original calculation assume the local ring is
𝑅 𝑝
a DVR, [Art82a, Prop 3.6] does work for any regular local rings [Mae97, Thm 2.1]. If 𝑔𝑝 is

part of a regular system of parameters of 𝑅𝑝, then sections 4 of [Art82a] tells us 𝑉Λ 𝑝 ⊗ 𝑅𝑝 is

indeed regular. If 𝑝 is a singular point of 𝑆1 or 𝑆2 which doesn’t lie in 𝑆1 ∩ 𝑆2), 𝑉Λ 𝑝 ⊗ 𝑅𝑝

is not regular. However, from the above equations, a direct calculations show that on each

31

standard affine chart (e.g. {𝜉11 = 𝜉21 = 𝜉31 = 1}), 𝑉Λ 𝑝 ⊗ 𝑅𝑝 can be defined by 4 equations.

Hence 𝑉Λ 𝑝 ⊗ 𝑅𝑝 has an open cover with each a complete intersection in A6

𝑅 𝑝 , furthermore
the coordinate ring of each affine chart is again a complete intersection as a C-algebra by

counting dimensions. Hence 𝑉Λ 𝑝 ⊗ 𝑅𝑝 is Cohen-Macaulay.

Consider the points (not necessarily closed) 𝑞 ∈ Spec(𝑅𝑝). If 𝑔𝑝 ∈ 𝑚𝑞, the fiber over 𝑞 is

the union of three standard Hirzebruch surfaces F1, meeting transversally, such that any pair

of them meet along a fiber of one and the (−1)-curve of the other ([Art82a, Prop. 3.10]). If

to P2

𝑔𝑝 ∉ 𝑚𝑞, the fiber over 𝑞 is completely determined by [𝜉11 : 𝜉12 : 𝜉13], hence is isomorphic
𝑅 𝑝 . So, in particular, the closed fiber is the union of three F1 as desired and all fibers
have same relative dimension. Again by the "Miracle flatness" theorem, 𝑉Λ 𝑝 ⊗ 𝑅𝑝 is flat over

Spec(𝑅𝑝). As 𝜏 is faithfully flat, 𝜋 𝑝 is flat in this case.

Case 3: 𝑝 ∈ 𝑆1 ∩ 𝑆2 or 𝑝 ∈ 𝑆1 ∩ {𝑥3
2

− 𝑥3

as in [Mae97, Prop. (2.2),Lemma (2.3)] shows that each fiber over Spec(𝒪P3

3 = 0}. We again write Λ𝑝 = ( 𝑓𝑝, 𝑔𝑝)𝜔. A same calculation
, 𝑝) has the

C

same relative dimension and the closed fiber is a cone over a twisted cubic as described in

Definition 1.0.6. Furthermore, in [Mae97, Lemma 2.4], Maeda shows the following facts:

a) 𝑉Λ 𝑝 has an open affine cover

𝑉Λ 𝑝 = 𝑈1 ∪ 𝑈2 ∪ 𝑈3

where 𝑈1 and 𝑈2 are hypersurfaces in A3

𝒪P3
C

.

, 𝑝

b) 𝑈3 is a (3, 3)− complete intersection in A4

𝒪P3
C

.

, 𝑝

Notice that here 𝑈1, 𝑈2, 𝑈3 do not have to be regular as 𝑓𝑝, 𝑔𝑝 are not part of a local parameters

in the maximal ideal of the local ring at 𝑝, for some 𝑝. For example, when 𝑝 = [0 : 0 : 1 : 0].

However, we can still conclude that 𝑈1, 𝑈2, 𝑈3 are Cohen-Macaulay since they are complete

intersection, hence so is 𝑉Λ 𝑝 . So we have 𝜋 𝑝 is flat by the "Miracle flatness" theorem.

32

As the base field is C, the argument above shows that the fiber over each geometric point is indeed

one of the three cases in Definition 1.0.6. This shows that 𝑉Λ is a Brauer-Severi surface bundle

over P3

C. We denote it by 𝑌 in the following sections of this paper as before.

□

Now we explain which surfaces in P3

C admit an associated Brauer-Severi surface bundle:

Definition 4.2.5. Let 𝑆 be a reduced surface in P3

C with irreducible components 𝑆 = 𝑆1∪𝑆2∪· · ·∪𝑆𝑚.
Then we say 𝑆 admits a nontrivial triple cover étale in codimension 1 if there is nontrivial element

in

𝑛
(cid:202)

𝑖=1

𝐻1(C(𝑆𝑖), Z/3) ∩ ker(

(cid:202)

(𝜕1

𝐶))

𝐶

Where 𝐶 runs over all irreducible curves in P3, 𝜕1

𝐶 is the residue map as in Definition 2.3.1.

It is clear that any surface admits a nontrivial triple cover étale in codimension 1 will give us a

3-torsion Brauer class in C(P3

C) by Bloch-Ogus sequence as discussed in Example 4.1.1.

So with the proof of Theorem 4.2.1, we have:

Corollary 4.2.6. Let 𝑆 ⊂ P3

C be a reduced surface which admits a nontrivial triple cover étale

in codimension 1 (Definition 4.2.5). Assume the 3-torsion Brauer class given by the Bloch-Ogus

sequence is represented by a cyclic algebra 𝒜 of degree 3. Then there exists a Brauer-Severi

surface bundle 𝑌𝑆 → P3

C with discriminant locus 𝑆 associated to 𝒜. Furthermore, 𝑌𝑆 is integral.

Proof. The first part of this corollary directly follows from a similar discussion of local structures

as in Theorem 4.2.1. Next, we show that 𝑌𝑆 is reduced. Indeed, the map 𝑌𝑆 → P3

C is projective,

hence closed. Then for any point 𝑦 ∈ 𝑌𝑆, there is a point 𝑦′ lying in a closed fiber such that 𝑦

specializes to 𝑦′. Since any localization of a reduced ring is again reduced, it suffices to check

the local ring 𝒪𝑌𝑆,𝑦′ is reduced. Further more it suffices to assume 𝑦′ is a closed point. This can

be directly checked using the explicit equations given in the proof of Theorem 4.2.1. (Details are

discussed in Lemma .0.6.)

33

On the other hand, 𝑌𝑆 is irreducible because P3

that there exists a dense collection of points in P3

C is irreducible, 𝜋 is flat hence open and the fact
C whose fiber is irreducible ([sta24a]). Hence 𝑌𝑆

is integral.

□

34

CHAPTER 5

THE SPECIALIZATION METHOD AND DESINGULARIZATION

5.1 The specialization method

In this section, we review the developments of the specialization methods. More details can be

found in [Pir18], [Tsc20] and [CT19].

This idea was firstly introduced by Clemens in [Cle75], a modern version of his theorem is

summarized as follows:

Theorem 5.1.1 ([Cle75],[Bea77],[Tsc20, Thm. 1]). Let 𝜙 : 𝒳 → 𝐵 be a flat family of projective

threefolds with smooth generic fiber. Assume that there exists a point 𝑏 ∈ 𝐵 such that the fiber

𝑋 := 𝒳𝑏 satisfies the following conditions:

• (S) Singularities: 𝑋 has at most rational double points.

• (O) Obstruction: the intermediate Jacobian of a desingularization ˜𝑋 of 𝑋 is not a product of

Jacobians of curves.

Then there exists a Zariski open subset 𝐵◦ ⊂ 𝐵 such that for all 𝑏′ ∈ 𝐵◦, the fiber 𝒳𝑏′ is not rational.

For the definition and first properties of the intermediate Jacobian, we refer to [CG72]. This

idea was further developed and modified within last 15 years. A novel idea of obstruction to stable

rationality appears firstly to Voisin in [Voi15]:

Definition 5.1.1. Let 𝑋 be a projective variety of dimension 𝑛 over a field 𝑘, assume 𝑋 (𝑘) ≠ 0.

Let [Δ𝑋] ∈ CH𝑛 (𝑋 ×𝑘 𝑋) be the diagonal Chow class, namely Δ𝑋 = {𝑥, 𝑥} ⊂ 𝑋 ×𝑘 𝑋. We say 𝑋

has integral Chow decomposition of the diagonal, if

[Δ𝑋] = [𝑋 × 𝑥] + [𝑍] ∈ CH𝑛 (𝑋 ×𝑘 𝑋).

Here 𝑥 is a 𝑘-point and 𝑍 is a 𝑛-cycle on 𝑋 ×𝑘 𝑋 whose support has form 𝐷 ×𝑘 𝑋 with 𝐷 a closed

subvariety of 𝑋 has codimension at least 1.

35

If 𝑋 does not admit an integral Chow decomposition of the diagonal, then 𝑋 is not stably

rational ([Tsc20]). Hence, the result in [Voi15] can be summarized as:

Theorem 5.1.2 ([Voi15, Thm .2.1],[Tsc20, Thm. 2]). Let 𝜙 : 𝒳 → 𝐵 be a flat family of projective

varieties with smooth generic fiber. Assume that there exists a point 𝑏 ∈ 𝐵 such that the fiber

𝑋 := 𝒳𝑏 satisfies the following conditions:

• (S) Singularities: 𝑋 has at most rational double points.

• (O) Obstruction: there exist a desingularization ˜𝑋 of 𝑋 which does not admit an integral

Chow decomposition of the diagonal.

Then a very general (which means the complement of a countable union of Zariski closed subsets

of 𝐵) fiber of 𝜙 does not admit an integral decomposition of the diagonal, and in particular, is not

stably rational.

In the literature, the special fiber as in Theorem 5.1.1 and Theorem 5.1.2 is called a reference

variety. Further developments of specialization method are trying to allow other type of singular-

ities and find practically useful obstructions of reference variety. In the present thesis we focus on

the refinement given in [CTP16] and [HPT18]:

Theorem 5.1.3 ([HPT18, Thm. 4]). Let 𝜙 : 𝒳 → 𝐵 be a flat family of projective complex varieties

with smooth generic fiber. Assume that there exists a point 𝑏 ∈ 𝐵 such that the fiber 𝑋 := 𝒳𝑏 is

integral and satisfies the following conditions:

• (S) Singularities: 𝑋 admits an universally CH0-trivial desingularization (Definition 1.0.3).

• (O) Obstruction: The unramified Brauer group (Definition 2.3.2) of 𝑋 is nontrivial.

Then a very general fiber of 𝜙 is not stably rational.

Theorem 5.1.3 is widely used in practice. However, one still need to construct a resolution of the

reference variety, which can be very hard for high dimension varieties. In [Sch19a, Proposition 26]

36

and [Sch19b], Schreieder provide a new refinement which only involve a purely cohomological

criteria:

Theorem 5.1.4 ([Sch19a, Proposition 26]). Let 𝜙 : 𝒳 → 𝐵 be a flat family of projective complex

varieties with smooth generic fiber. Assume that there exists a point 𝑏 ∈ 𝐵 such that the fiber

𝑋 := 𝒳𝑏 is integral satisfies the following conditions:

• (O) Obstruction: The unramified Brauer group of 𝑋 is nontrivial.

• (S) Singularities: Suppose there exists a resolution of singularities 𝜏 : ˜𝑋 → 𝑋, let 𝑈 be the

smooth locus of 𝑋, and 𝛼 be a nontrivial unramified Brauer class. We require the restriction

of 𝛼 to the function field of each irreducible component of ˜𝑋 − 𝜏−1(𝑈) is trivial (See Lemma

5.2.1).

Then a very general fiber of 𝜙 is not stably rational.

Note that Schreieder’s result can be further generalized by replacing resolutions of singularities

by alterations ([Sch19b]). In a very recent paper, Pirutka modified Schreieder’s idea in [Pir23].

By introducing the notation of relative unramified Brauer group of a fibration, one can check the

singularity requirements by passing to the completion of local rings.

5.2 Desingularization

In this section, we show that Example 4.1.1 is a reference variety by applying Theorem 5.1.4.

Lemma 5.2.1 ([Sch21a, Proposition 4.8(a)]). Let 𝑌 be a projective variety over a field 𝑘. Let 𝐸 ⊂ 𝑌

be an irreducible subvariety such that the local ring of 𝑌 at the generic point 𝜂𝐸 of 𝐸, denoted by

𝒪𝑌 ,𝜂𝐸 , is a regular local ring. Then there exists a restriction map:

Res𝑌

𝐸 : 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑘, Z/3) → 𝐻2(𝑘 (𝐸), Z/3)

Proof. Let 𝛼 ∈ 𝐻2

𝑛𝑟 (𝑘 (𝑌 )/𝑘, Z/3) be an unramified Brauer class (Definition 2.3.2). Notice that by

assumption, 𝒪𝑌 ,𝜂𝐸 is a regular local ring with residue field 𝑘 (𝐸) and fraction field 𝑘 (𝑌 ). We have

37

the following diagram:

0

𝐻2(𝒪𝑌 ,𝜂𝐸 , Z/3)

𝐻2(𝑘 (𝐸), Z/3)

𝐻2(𝑘 (𝑌 ), Z/3)

⊕𝜕2
𝜈

(cid:201) 𝐻1(𝑘 (𝜈), Z/3)

here the left column is given by [CT95, Theorem 3.8.3]. The horizontal map is given by the

functoriality in étale cohomology. Now that 𝛼 is an unramified Brauer class, it is killed by ⊕𝜕2
𝜈 .

Hence 𝛼 comes from a class in 𝐻2(𝒪𝑌 ,𝜂𝐸 , Z/3), which can be further mapped to 𝐻2(𝑘 (𝐸), Z/3) by

the horizontal map.

□

We use notation in Example 4.1.1 and Lemma 5.2.1. Let 𝑈 ⊂ 𝑌 be the smooth locus of 𝑌 .

Let 𝛼1 ∈ 𝐵𝑟 (C(P3

C)) [3] be the Brauer class which has nontrivial residue 𝛾1 along 𝑆1, and trivial

residues everywhere else. By Lemma 2.3.1, 𝛼1 can be represents by the cyclic algebra

𝑥3
2

(

− 𝑥3
3
𝑥3
0

,

𝐹𝑆1
𝑥9
0

)𝜔.

By arguments in Example 4.1.1, 𝛼1 can be lifted to a nontrivial unramified Brauer class

˜𝛼1 ∈ 𝐻2

𝑛𝑟 (C(𝑌 )/C, Z/3)

Now we prove that the second hypothesis in [Sch19a, Proposition 26] is true in our case:

Lemma 5.2.2. Let 𝜋 : 𝑌 → P3

C be the Brauer-Severi surface bundle in Example 4.1.1. Let 𝑈 be
the smooth locus of 𝑌 . Then there exists a resolution of singularities 𝑓 : ˜𝑌 → 𝑌 , such that for each

irreducible component 𝐸 of ˜𝑌 − 𝑓 −1(𝑈), Res ˜𝑌

𝐸 ( ˜𝛼1) is trivial.

Proof. The existence of resolution of singularities of 𝑌 is guaranteed by Hironaka’s theorem

[Hir64]. We can further assume without loss of generality that each 𝐸 is a prime divisor of ˜𝑌 .

Recall that 𝑌 has singular locus of codimension at least 2. So for every irreducible component

𝐸 of ˜𝑌 − 𝑓 −1(𝑈), 𝑓 (𝐸) has dimension at most 3. On the other hand, since that the generic fiber of 𝜋

38

is smooth, and the generic fiber over each irreducible component of the discriminant locus (namely

𝑆1 ∪ 𝑆2) is the union of three standard Hirzebruch surfaces F1 described in Definition 1.0.6. We

know 𝜋( 𝑓 (𝐸)) has dimension at most 1 (This follows from that local model of 𝜋 over 𝑆1 and 𝑆2 is

smooth, see the discussion in Theorem 4.2.1). In other words, each 𝐸 in ˜𝑌 would dominate a curve

or a point in 𝑆1 ∪ 𝑆2.

In the following of this proof, let 𝐾 = C(P3

C) be the function field of P3

C. Denote by 𝑝𝐸 the

generic point of 𝜋( 𝑓 (𝐸)), by 𝐾𝑃𝐸 the field of fractions of the regular complete local ring ˆ𝒪P3

C,𝑝𝐸 .

We give a case by case argument according to the generic point 𝑝𝐸 ∈ P3

C of 𝜋( 𝑓 (𝐸)):

Case 1: 𝑝𝐸 ∉ 𝑆1. Then Res ˜𝑌

𝐸 ( ˜𝛼1) = 0 simply follows from the fact that 𝑝𝐸 does not belongs to the

discriminant locus of 𝛼1 (see e.g. the proof of [Sch19b, Proposition 5.1(2)]).

Case 2: 𝑝𝐸 ∈ 𝑆1 − 𝑆2. Consider the following commutative diagram coming from functoriality:

𝐻2(𝐾 (𝑌𝐾), Z/3Z)

𝐻2(𝐾𝑝𝐸 (𝑌𝐾), Z/3Z)

𝐻2(𝐾, Z/3Z)

𝐻2(𝐾𝑝𝐸 , Z/3Z)

Because ˜𝛼1 is unramified, by [Pir23, Proposition 2.5], it suffices to check 𝛼1 = 0 in

𝐻2(𝐾𝑝𝐸 (𝑌𝐾), Z/3Z).
−𝑥3
3

algebra 𝒜 = (

𝐹𝑆
2

(𝑥3
2
𝑥21
0

Indeed, as 𝑌𝐾 is the Brauer-Severi surface associate to the cyclic
, 𝐹𝑆
1
𝑥9
0

)𝜔 and 𝐹𝑆2 is a nonzero cube when 𝐹𝑆1 = 0 in the residue field of

)

𝒪P3

C,𝑝𝐸 . By Cohen’s structure theorem [Coh46, Theorem 15], the residue field embeds into

𝐾𝑃𝐸 . Hence, after taking base changes to 𝐾𝑝𝐸 ,

(

− 𝑥3
3

𝐹𝑆2 (𝑥3
2
𝑥21
0

)

,

𝐹𝑆1
𝑥9
0

)𝜔 (cid:27) (

𝑥3
2

− 𝑥3
3
𝑥3
0

,

𝐹𝑆1
𝑥9
0

)𝜔.

This implies that 𝑌𝐾 is also the Brauer-Severi surface associate to 𝛼1, we conclude that 𝛼1 = 0

in 𝐻2(𝐾𝑝𝐸 (𝑌𝐾), Z/3Z) by Amitsur’s theorem [GS06, Theorem 5.4.1].

Case 3: 𝑝𝐸 ∈ 𝑆1 ∩ 𝑆2 is a closed point, and 𝑝𝐸 ∉ {𝑥3
2

− 𝑥3

3 = 0}. Then notice 𝛼1 can be represented

by the cyclic algebra

𝑥3
2

(

− 𝑥3
3
𝑥3
0

,

𝐹𝑆1
𝑥9
0

)𝜔 (cid:27) (

(𝑥2 − 𝑥3)6
− 𝑥3
(𝑥3
)2
3
2

,

𝐹𝑆1
− 𝑥3
3

)3

(𝑥3
2

)𝜔.

39

Then (𝑥2−𝑥3)6
(𝑥3
−𝑥3
2
3

C,𝑝𝐸 , which is C. Hence (𝑥2−𝑥3)6

)2 is nonzero in the residue field of 𝒪P3

)2 is a cube
in the residue field as C is algebraically closed. By Cohen’s structure theorem, the residue
field C embeds into 𝐾𝑃𝐸 , hence (𝑥2−𝑥3)6
)2 is also a cube in 𝐾𝑃𝐸 . This shows that 𝛼1 is trivial
−𝑥3
3
in 𝐻2(𝐾𝑃𝐸 , Z/3Z). By the same commutative diagram as in case 2, it is clear that 𝛼1 = 0 in
𝐻2(𝐾𝑝𝐸 (𝑌𝐾), Z/3Z). We then have Res ˜𝑌

𝐸 ( ˜𝛼1) = 0 by [Pir23, Proposition 2.5].

−𝑥3
3

(𝑥3
2

(𝑥3
2

Case 4: 𝑝𝐸 is one of [0 : 0 : 1 : 1],[0 : 0 : 1 : 𝜔],[0 : 0 : 1 : 𝜔2]. In these cases, by the discussions

in Case 4 of the proof of Theorem 3.3.1, we can choose another appropriate representing

algebras of 𝛼1:

𝑥6
2
) (𝑥3
3
It is straight forward to check this is a representing algebra of 𝛼1 by applying Lemma 2.3.1.
C,𝑝𝐸 , which is C. Hence the

) is a nontrivial unit in the residue field of 𝒪P3

𝐹𝑆1
𝑥9
0

− 𝑥3
1

− 𝑥3
2

(𝑥3
1

And

)𝜔.

(

)

,

𝑥6
2
)(𝑥3
3

−𝑥3
1

(𝑥3
1

−𝑥3
2

remaining proof can be done exactly same as in Case 3.

Case 5: 𝑝𝐸 = [0 : 1 : 0 : 0]. In this case, the proof are the same as in Case 4, by using the following

representing algebra of 𝛼1:

− 𝑥3
2
Case 6: 𝑝𝐸 is one of the generic point of 𝐷1, 𝐷2 and 𝐷3(Example 4.1.1). Note that by the defining

− 𝑥3
1

(𝑥3
1

)

(

𝑥6
1
) (𝑥3
3

,

𝐹𝑆1
𝑥9
0

)𝜔.

equations of 𝐷1, 𝐷2 and 𝐷3,

−𝑥3
𝑥3
2
3
𝑥3
0

is always a nontrivial cube in the residue field of 𝒪P3

C,𝑝𝐸 .

Again as the residue field embeds into 𝐾𝑝𝐸 , we get 𝛼1 = 0 in 𝐻2(𝐾𝑝𝐸 (𝑌𝐾), Z/3Z).

This completes the proof.

5.3 Main result

With Example 4.1.1, we prove Theorem 1.0.8:

□

Theorem 1.0.8. There exists a flat projective family of Brauer-Severi surface bundles over P3
C,

where a general fiber in this family is smooth and not stably rational.

40

Proof. By [Sch19a, Proposition 26] and Lemma 5.2.2,we know the Brauer-Severi surface bundle

constructed in Example 4.1.1 can be used as a reference variety.

To finish the proof, we need to construct a flat family of Brauer-Severi surface bundles over

P3
C with Example 4.1.1 as one closed fiber with smooth general fiber. Start with the cyclic algebra

from Example 4.1.1:

𝒜 = (

− 𝑥3
3

𝐹𝑆2 (𝑥3
2
𝑥21
0

)

,

𝐹𝑆1
𝑥9
0

)𝜔

We consider two regular surfaces in P3
C:

𝐺1 = {𝑥9

0 − 𝑥9

1 + 𝑥8

2

𝑥3 + 𝑥8
3

𝑥2 = 0}

𝐺2 = {𝑥21

0 + 𝑥21

1 + 𝑥21

2 − 𝑥21

3 = 0}

By Lemma .0.3, both 𝐺1 and 𝐺2 are regular surfaces in P3

C, and they intersect transversally.

Consider the following pencil of cyclic algebras:

𝒜[𝑡0:𝑡1] = (

𝑡0𝐹𝑆2 (𝑥3
2

− 𝑥3
3

) + 𝑡1(𝐺2 − 𝐹𝑆2 (𝑥3
2

− 𝑥3
3

))

𝑥21
0

,

𝑡0𝐹𝑆1 + 𝑡1(𝐺1 − 𝐹𝑆1)
𝑥9
0

)𝜔

We denote

and

𝑡0𝐹𝑆2 (𝑥3

2 − 𝑥3

3) + 𝑡1(𝐺2 − 𝐹𝑆2 (𝑥3

2 − 𝑥3

3))

𝑡0𝐹𝑆1 + 𝑡1(𝐺1 − 𝐹𝑆1)

by 𝐹 [𝑡0:𝑡1]
𝑆2
𝐹 [𝑡0:𝑡1]
𝑆2

and 𝐹 [𝑡0:𝑡1]
𝑆1
= 0 and 𝐹 [𝑡0:𝑡1]

𝑆1

respectively. By Lemma .0.4 and Lemma .0.5, when [𝑡0 : 𝑡1] ≠ [1 : 0], both

= 0 are irreducible surfaces in P3. Using Lemma 2.3.1, the induced triple

covers are given by

𝛾 [𝑡0:𝑡1]
1

=

𝐹 [𝑡0:𝑡1]
𝑆2
𝑥21
0

, 𝛾 [𝑡0:𝑡1]
2

=

𝑥9
0
𝐹 [𝑡0:𝑡1]
𝑆1

Similar to the discussion in Example 4.1.1, note that the residue of 𝛾 [𝑡0:𝑡1]
at the point [0 : 𝜉 : 1 : 1] is 1 ∈ Z/3, where 𝜉 satisfies 𝜉9 − 2 = 0. And the residue of 𝛾 [𝑡0:𝑡1]

of the valuation centered

of

1

2

the valuation centered at the point [0 : 𝜓 : 1 : −1] is 2 ∈ Z/3, where 𝜓21 + 2 = 0. Hence both

41

and 𝛾 [𝑡0:𝑡1]
2

𝛾 [𝑡0:𝑡1]
1
Brauer-Severi surface bundle 𝒴[𝑡0:𝑡1] → P3
C.

are irreducible. By Corollary 4.2, for any [𝑡0 : 𝑡1] ∈ P1

C, there exists an integral

By viewing 𝒜[𝑡0:𝑡1] as a simple algebra over P1

C × P3

C and applying the construction of Theorem

4.2.1 again, we have a Brauer-Severi surface bundle over P1

C which can be viewed as a 1

dimensional family of of Brauer-Severi surface bundles over P3

C denote this family of

C × P3
C. Let 𝒴 → P1

Brauer-Severi surface bundles. We claim this is indeed a flat family. By [Har77, Proposition 9.7],

It is sufficient to check 𝒴 is integral. 𝒴 is irreducible because each closed fiber is an irreducible

variety and the morphism 𝒴 → P1

C is projective, hence closed. Now let ˆ𝒴 be the closed sub-scheme

of 𝒴 with the same underlying topological space equipped with the reduced scheme structure, we

have the following Cartesian diagram:

ˆ𝒴[𝑡0:𝑡1]
𝑖 [𝑡
0:𝑡
𝒴[𝑡0:𝑡1]

1 ]

{[𝑡0 : 𝑡1]}

ˆ𝒴
𝑖

𝒴

P1
C

On one hand, 𝑖 [𝑡0:𝑡1] is a homeomorphism on topological spaces as the pullback of schemes by

monomorphism coincide with topological pullback according to the explicit construction of fiber

product of ringed spaces. On the other hand, 𝑖 [𝑡0:𝑡1] is a closed immersion as a base change of the
closed immersion 𝑖. Since 𝒴[𝑡0:𝑡1] is reduced as discussed above, we know 𝑖 [𝑡0:𝑡1] is an isomorphism.
Finally, by replacing 𝒴 by ˆ𝒴 if necessary, we get a 1 dimensional flat family of Brauer-Severi

surface bundles over P3

C, with a special fiber 𝒴[1:0] (Example 4.1.1 and Lemma 5.2.2) and a regular

fiber 𝒴[1:1] ([Mae97, Theorem 2.1]), hence we are done.

□

42

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The stacks project. https://stacks.math.columbia.edu/tag/00R4, 2023.

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The stacks project. https://stacks.math.columbia.edu/tag/004Z, 2024.

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The stacks project. https://stacks.math.columbia.edu/tag/031R, 2024.

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ADAM TOPAZ. Abelian-by-central galois groups of fields i: A formal description.
Transactions of the American Mathematical Society, 369(4):pp. 2721–2745, 2017.

Yuri Tschinkel. Rationality and specialization. Afrika Matematika, 31(1):191–205,
2020.

Michel Van den Bergh. The Brauer-Severi scheme of the trace ring of generic
matrices. In Perspectives in ring theory (Antwerp, 1987), volume 233 of NATO Adv.
Sci. Inst. Ser. C: Math. Phys. Sci., pages 333–338. Kluwer Acad. Publ., Dordrecht,
1988.

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Claire Voisin. Unirational threefolds with no universal codimension 2 cycle. Invent.
Math., 201(1):207–237, 2015.

45

APPENDIX

Lemma .0.1. Let 𝑃1 = 𝑃1(𝑥1, 𝑥2, 𝑥3) = (𝑥3
1

) (𝑥3
2
Then for any [𝑡1 : 𝑡2] ≠ [0 : 1] or [1 : 0] or [1 :

−𝑥3
2

−𝑥3
3
√

) (𝑥3
3

−𝑥3
1

), let 𝑃2 = 𝑃2(𝑥1, 𝑥2, 𝑥3) = 𝑥3
1

𝑥3
3.
−27], singular locus of the

𝑥3
2

√

−27] or [1 : −

curve 𝑡1𝑃1 + 𝑡2𝑃2 = 0 in P2

C = Proj C[𝑥1, 𝑥2, 𝑥3] are precisely the three points:

[1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1]

Proof. Taking partial derivatives:

𝑡1

𝑡1

𝑡1

𝜕𝑃1
𝜕𝑥1
𝜕𝑃1
𝜕𝑥2
𝜕𝑃1
𝜕𝑥3

+ 𝑡2

+ 𝑡2

+ 𝑡2

𝜕𝑃2
𝜕𝑥1
𝜕𝑃2
𝜕𝑥2
𝜕𝑃2
𝜕𝑥3

= 0

= 0

= 0

Consider a point with 𝑥1 = 0 lying in the curve. Then 𝑃2 = 0, and this forces 𝑃1 = 𝑥3
2

𝑥3
3

(𝑥3
3

−𝑥3
2

) = 0

as 𝑡1 ≠ 0. If one of 𝑥2 or 𝑥3 is 0, then we get the singularities listed in the statement of this lemma.
3 = 0, and the above partial derivatives can be simplified to obtain 𝑥2 = 0,
and hence 𝑥3 = 0, which is a contradiction. Similar calculations work when 𝑥2 = 0 or 𝑥3 = 0. So

If not, we have 𝑥3
2

− 𝑥3

all singularities when some 𝑥𝑖 equal to 0 are precisely the three points listed above.

In the following we assume all of 𝑥1, 𝑥2 and 𝑥3 are nonzero. It is clear that for 𝑖 ≠ 𝑗, we have
𝜕𝑃2
𝜕𝑥𝑖

𝜕𝑃2
𝜕𝑥 𝑗 . The above partial derivatives also tell us

= 𝑥 𝑗

𝑥𝑖

𝑥1

𝜕𝑃1
𝜕𝑥1

= 𝑥2

𝜕𝑃1
𝜕𝑥2

= 𝑥3

𝜕𝑃1
𝜕𝑥3

.

By Euler’s theorem on homogeneous polynomial, we know

9𝑃1 = 𝑥1

𝜕𝑃1
𝜕𝑥1

+ 𝑥2

𝜕𝑃1
𝜕𝑥2

+ 𝑥3

𝜕𝑃1
𝜕𝑥3

.

𝑥1

𝜕𝑃1
𝜕𝑥1

= 3𝑃1,

Hence we have

which simplifies to

(𝑥3

2 − 𝑥3

3) (𝑥3

2

3 − 𝑥6
𝑥3

1) = 0

46

Similarly,

(𝑥3

3 − 𝑥3

1) (𝑥3

1

(𝑥3

1 − 𝑥3

2) (𝑥3

1

3 − 𝑥6
𝑥3

2) = 0

𝑥3
2 − 𝑥6

3) = 0

By our assumption here, 𝑃2 ≠ 0, so 𝑃1 ≠ 0. Hence 𝑥3

𝑖 ≠ 𝑥3

𝑗 , so we have

𝑥3
2

𝑥3
3 − 𝑥6

1 = 0

𝑥3
1

𝑥3
3 − 𝑥6

2 = 0

𝑥3
1

𝑥3
2 − 𝑥6

3 = 0

Hence the original partial derivatives simplify to

3𝑡1(𝑥3

3 − 𝑥3

2) + 𝑡2𝑥3

1 = 0

3𝑡1(𝑥3

2 − 𝑥3

1) + 𝑡2𝑥3

3 = 0

3𝑡1(𝑥3

1 − 𝑥3

3) + 𝑡2𝑥3

2 = 0

Viewing this as three linear equations of 𝑥3

matrix as −𝑡2(27𝑡2

1 + 𝑡2
determinant is non-zero, we know 𝑥3

𝑖 , we can calculate the determinant of the coefficient
2). Hence when [𝑡1 : 𝑡2] is not one of the four cases listed in the statement, the
□

0 = 0 is the only solution, which is impossible.

1 = 𝑥3

2 = 𝑥3

Lemma .0.2. Using notations in Example 4.1.1, singular locus of 𝑆2 consists of three lines:

𝐿1 = {𝑥0 = 0, 𝑥1 = 0}

𝐿2 = {𝑥0 = 0, 𝑥2 = 0}

𝐿3 = {𝑥0 = 0, 𝑥3 = 0}

Each 𝐷𝑖 exactly passes through 5 singular points of 𝑆2, they are as follows:

𝐷1 : [0 : 0 : 0 : 1], [0 : 0 : 1 : 0], [0 : 0 : 1 : 1], [0 : 0 : 1 : 𝜔], [0 : 0 : 1 : 𝜔2]

𝐷2 : [0 : 0 : 0 : 1], [0 : 1 : 0 : 0], [0 : 1 : 0 : 1], [0 : 1 : 0 : 𝜔], [0 : 1 : 0 : 𝜔2]

𝐷3 : [0 : 0 : 1 : 0], [0 : 1 : 0 : 0], [0 : 1 : 1 : 0], [0 : 1 : 𝜔 : 0], [0 : 1 : 𝜔2 : 0]

47

Proof. Let 𝑃1 = 𝑃1(𝑥1, 𝑥2, 𝑥3) = (𝑥3
1

− 𝑥3
2
Then any singular point of 𝑆2 satisfies the one of the two systems of equations:

− 𝑥3
3

− 𝑥3
1

) (𝑥3
2

) (𝑥3
3

), and let 𝑃2 = 𝑃2(𝑥1, 𝑥2, 𝑥3) = 𝑥3
1

𝑥3
2

𝑥3
3.

𝑥0 = 0

𝑃2

𝜕𝑃1
𝜕𝑥1

𝑃2

𝜕𝑃1
𝜕𝑥2

𝑃2

𝜕𝑃1
𝜕𝑥3

+ (𝑃1 − 2𝑃2) 𝜕𝑃2
𝜕𝑥1
+ (𝑃1 − 2𝑃2) 𝜕𝑃2
𝜕𝑥2
+ (𝑃1 − 2𝑃2) 𝜕𝑃2
𝜕𝑥3

= 0

= 0

= 0

.






or

2 (𝑃1 − 𝑃2)

𝑥9
0 = − 1
(𝑃1 + 𝑃2) 𝜕𝑃1
𝜕𝑥1
(𝑃1 + 𝑃2) 𝜕𝑃1
𝜕𝑥2
(𝑃1 + 𝑃2) 𝜕𝑃1
𝜕𝑥3






+ (𝑃1 − 3𝑃2) 𝜕𝑃2
𝜕𝑥1
+ (𝑃1 − 3𝑃2) 𝜕𝑃2
𝜕𝑥2
+ (𝑃1 − 3𝑃2) 𝜕𝑃2
𝜕𝑥3

= 0

= 0

= 0

.

(.0.1)

(.0.2)

In case (A.1), clearly points in 𝐿1 ∪ 𝐿2 ∪ 𝐿3 are solutions of this system of equations. Assume

𝑥𝑖 ≠ 0, 𝑖 = 1, 2, 3. Then multiply the second equation of (A.1) by 𝑥1, multiply the third equation of

(A.1) by 𝑥2, multiply the last equation of (A.1) by 𝑥3 and add the resulting equations. By Euler’s

theorem on homogeneous polynomial, we have

𝑃1 − 𝑃2 = 0

and the partial derivatives can be simplified as

𝑥0 = 0

𝜕𝑃1
𝜕𝑥1
𝜕𝑃1
𝜕𝑥2
𝜕𝑃1
𝜕𝑥3

−

−

−

𝜕𝑃2
𝜕𝑥1
𝜕𝑃2
𝜕𝑥2
𝜕𝑃2
𝜕𝑥3

= 0

= 0

= 0

Hence by Lemma .0.1, the set of all singularities in this case is identified to 𝐿1 ∪ 𝐿2 ∪ 𝐿3.

48

In case (A.2), it is easy to check if some 𝑥𝑖 = 0, 𝑖 = 1, 2, 3, then either we are reduced to case

(A.1) or we obtain that all of them are 0. So we again assume 𝑥𝑖 ≠ 0, 𝑖 = 1, 2, 3, and use the same

trick as the previous case. By Euler’s theorem on homogeneous polynomial, we have

0 = 𝑃2

1 + 2𝑃1𝑃2 − 3𝑃2

2 = (𝑃1 − 𝑃2) (𝑃1 + 3𝑃2)

If 𝑃1 − 𝑃2 = 0, then 𝑥0 = 0, we are reduced to the case (A.1). So the only new possibility

is 𝑃1 + 3𝑃2 = 0. Then the partial derivatives are exactly the partial derivatives for the curve

𝑃1 + 3𝑃2 = 0. Again by Lemma .0.1, the set of singularities of 𝑆2 are 𝐿1 ∪ 𝐿2 ∪ 𝐿3.

□

Lemma .0.3. Let

𝐺1 = {𝑥9

0 − 𝑥9

1 + 𝑥8

2

𝑥3 + 𝑥8
3

𝑥2 = 0}

𝐺2 = {𝑥21

0 + 𝑥21

1 + 𝑥21

2 − 𝑥21

3 = 0}

Then 𝐺1 and 𝐺2 are regular surface in P3

C. And furthermore 𝐺1 and 𝐺2 intersect transversally.

Proof. 𝐺2 is clearly a regular surface in P3

C. Taking partial derivatives of defining equation of 𝐺1,

the singular points are defined by the equations:

9𝑥8

0 = 0

−9𝑥8

1 = 0

8𝑥7
2

𝑥3 + 𝑥8

3 = 0

8𝑥7
3

𝑥2 + 𝑥8

2 = 0

This gives 𝑥0 = 𝑥1 = 0. Note that if one of 𝑥2 or 𝑥3 is 0, so is the other. Assume 𝑥2 ≠ 0 and 𝑥3 ≠ 0,
2 = 0. Then again 𝑥2 = 𝑥3 = 0, a contradiction. Hence 𝐺1 is also a

+ 𝑥7

+ 𝑥7

we get 8𝑥7
3 = 0, 8𝑥7
2
regular surface in P3
C.

3

To check 𝐺1 intersects 𝐺2 transversally, we prove by contradiction: Assume there is a point

𝑃 = [𝑥0 : 𝑥1 : 𝑥2 : 𝑥3] ∈ 𝐺1 ∩ 𝐺2, such that there exists a nonzero complex number 𝑘, with

21𝑥20

0 = 𝑘 (9𝑥8
0)

49

−21𝑥20

1 = 𝑘 (9𝑥8
1)

21𝑥20

2 = 𝑘 (8𝑥7

2

𝑥3 + 𝑥8
3)

−21𝑥20

3 = 𝑘 (8𝑥7

3

𝑥2 + 𝑥8
2)

Where the left side of each equations is the partial derivatives of defining equations of 𝐺1, and the

right hand side is 𝑘 times the partial derivatives of defining equations of 𝐺2. We split into several

cases:

Case 1: 𝑥2 = 0 or 𝑥3 = 0. In this case, we clearly have 𝑥2 = 𝑥3 = 0, hence 𝑥0 ≠ 0 and 𝑥1 ≠ 0. So the

partial derivatives with respect 𝑥0 and 𝑥1 tells us:

As 𝑃 ∈ 𝐺1 ∩ 𝐺2, we also have

𝑥12
0 =

9
21

𝑘 = −𝑥12
1

0 − 𝑥9
𝑥9

1 = 0

0 + 𝑥21
𝑥21

1 = 0

This forces 𝑥0 = 𝑥1 = 0, which makes this case impossible.

Case 2: 𝑥2 ≠ 0 and 𝑥3 ≠ 0. Then the partial derivatives with respect to 𝑥2 and 𝑥3 shows that

𝑥20
2
𝑥3 + 𝑥8
3

8𝑥7
2

=

𝑘

21

= −

𝑥20
3
𝑥2 + 𝑥8
2

8𝑥7
3

(note that the denominator is nonzero, otherwise by the partial derivatives with respect to 𝑥2

and 𝑥3, one of 𝑥2 or 𝑥3 is 0. This contradicts to the assumption.) This can be simplified to

𝑥21
2
+ 𝑥7
3

8𝑥7
2

= −

𝑥21
3
+ 𝑥7
2

8𝑥7
3

Since 𝑥3 ≠ 0, we assume 𝑥3 = 1 without lose of generality. Let 𝑡 = 𝑥7

2, we see the above

relation shows that 𝑡 is a root of the following polynomial:

Now we have three subcases:

𝑡4 + 8𝑡3 + 8𝑡 + 1 = 0

50

Sub-case 1: 𝑥0 = 0 and 𝑥1 = 0. Then the defining polynomial of 𝐺2 tells us

𝑡3 − 1 = 0

This contradicts to the relation: 𝑡4 + 8𝑡3 + 8𝑡 + 1 = 0.

Sub-case 2: 𝑥0 ≠ 0 and 𝑥1 ≠ 0. In this case, a similar argument as in Case 1 shows that

0 + 𝑥12
𝑥12

1 = 0

So we may assume 𝑥0 = 𝜏𝑥1, where 𝜏 satisfies 𝜏12 + 1 = 0. Plug these information into

defining equations of 𝐺1 and 𝐺2, we get:

𝐺1 : (𝜏9 − 1)𝑥9

1 + 𝑥2𝑥3(𝑡 + 1) = 0

𝐺2 : −(𝜏9 − 1)𝑥21

1 + 𝑡3 − 1 = 0

Note that 𝜏9 − 1 ≠ 0, and hence the above two equations shows that 𝑡 + 1 ≠ 0 and

𝑡3 − 1 ≠ 0. Taking ratio of the above two equations, we have:

−𝑥12

1 =

𝑡3 − 1
𝑥2𝑥3(𝑡 + 1)

Compare the above relation with 𝐺1, we have:

Hence we get:

This is simplified to

That is

𝑥3
1 =

(𝜏9 − 1) (𝑡3 − 1)
(𝑡 + 1)2

𝑥2
2

𝑡3 − 1
𝜏9 − 1

= 𝑥21

1 =

(𝜏9 − 1)7(𝑡3 − 1)7
𝑡2(𝑡 + 1)14

(𝜏9 − 1)8(𝑡3 − 1)6 − 𝑡2(𝑡 + 1)14 = 0

(𝜏9 − 1)8 =

𝑡2(𝑡 + 1)14
(𝑡3 − 1)6

Since 𝑡 satisfies 𝑡4 + 8𝑡3 + 8𝑡 + 1 = 0, we list all roots of this polynomial:

𝑡1 = −2 −

3
√

2

−

√︂

15
2

√

+ 6

2

51

𝑡2 = −2 −

𝑡3 = −2 +

3
√

2

√︂

15
2
√︂

+

3
√
2
√

−

−1

−

√
2

+ 6

√

+ 6

2

15
2

√

√︂

𝑡4 = −2 +

15
2
𝑡2 (𝑡+1)14
(𝑡3−1)6 for each value of 𝑡 above, we see
all four possible values of 𝑡 are impossible. (Indeed, one can check the norm of the left

By taking norm of both sides of (𝜏9 − 1)8 =

3
√
2

+ 6

−1

−

+

√
2

hand side has two possible estimated values: 0.118 or 135.882, while the norm of the

right hand side has two possible estimated values: 0.00238 or 14.2778.)

Sub-case 3: One of 𝑥0 or 𝑥1 is 0, and the other is nonzero. A similar discussion as in Sub-case 2

gives us

1 =

𝑡2(𝑡 + 1)14
(𝑡3 − 1)6

Again by taking norms of both sides, we see this is also impossible.

This completes the calculation.

□

Lemma .0.4. Use notations of Example 4.1.1 and Lemma .0.2, let

𝐺1 = {𝑥9

0 − 𝑥9

1 + 𝑥8

2

𝑥3 + 𝑥8
3

𝑥2 = 0}

Then for any [𝑡0 : 𝑡1] ≠ [1 : 0] ∈ P1
C,

𝐹 [𝑡0:𝑡1]
𝑆1

= 𝑡0𝐹𝑆1 + 𝑡1(𝐺1 − 𝐹𝑆1) = 0

defines irreducible surfaces in P3
C.

Proof. We can view 𝐹 [𝑡0:𝑡1]

𝑆1

as a polynomial in 𝑥0:

𝐹 [𝑡0:𝑡1]
𝑆1

= 𝑡0𝑥9

0 + 𝑡1(−𝑥9

1 + 𝑥8

2

𝑥3 + 𝑥8
3

𝑥2) + (𝑡0 − 𝑡1)𝑃1 = 0

Hence by the Eisenstein’s criterion, it suffices to show the curve in P2

C defined by the constant term

𝑡1(−𝑥9

1 + 𝑥8

2

𝑥3 + 𝑥8
3

𝑥2) + (𝑡0 − 𝑡1)𝑃1 = 0

has a regular point. It is clear that [0 : 1 : 0] is such a point.

□

52

Lemma .0.5. Use notations of Example 4.1.1 and Lemma .0.2, let

𝐺2 = {𝑥21

0 + 𝑥21

1 + 𝑥21

2 − 𝑥21

3 = 0}

Then for any [𝑡0 : 𝑡1] ≠ [1 : 0] ∈ P1
C,

𝐹 [𝑡0:𝑡1]
𝑆2

= 𝑡0𝐹𝑆2 (𝑥3

2 − 𝑥3

3) + 𝑡1(𝐺2 − 𝐹𝑆2 (𝑥3

2 − 𝑥3

3)) = 0

define irreducible surfaces in P3
C.

Proof. View 𝐹 [𝑡0:𝑡1]

𝑆2

as a polynomial of 𝑥0. As any factor of a homogeneous polynomial is also

homogeneous, it suffices to show the constant term with respect to 𝑥0 is itself irreducible. That is

we need to show that

𝑡1𝑥21

1 − (𝑡0 − 𝑡1)(𝑥3

2 − 𝑥3

3)𝑃2(𝑃1 − 𝑃2) + 𝑡1(𝑥21

2 − 𝑥21

3 ) = 0

is irreducible. View the above polynomial as a polynomial of 𝑥1, using Eisenstein criterion with

the prime factor (𝑥2 − 𝑥3), we get the conclusion.

□

Lemma .0.6. With notations as in the proof of Theorem 4.2.1 and Corollary 4.2.6, we have that 𝑌𝑆

is reduced.

Proof. As stated in the proof of Corollary 4.2.6, it suffices to check that over any closed point

𝑝 ∈ P3

C, and any point 𝑦 lying in the fiber over 𝑝, the local ring 𝒪𝑉Λ 𝑝 ,𝑦 (cid:27) 𝒪𝑌𝑆,𝑦 is reduced.

In the proof of Theorem 4.2.1, we provide open affine covers for each local model 𝑉Λ 𝑝 . Hence

it suffice to show the coordinate ring of each open affine set appearing in these open affine covers is

reduced. We discuss the cases as in the proof of Theorem 4.2.1 separately. First, Case 1 is trivial

as the local model is regular.

In Case 2, we consider the the affine chart {𝜉11 = 𝜉21 = 𝜉31 = 1}. Then its coordinate ring is

𝑅𝑝 [𝜉12, 𝜉13, 𝜉22, 𝜉23, 𝜉32, 𝜉33]/(𝑔𝑝𝜉22 − 𝜉12, 𝑔𝑝𝜉23 − 𝜉13, 𝑔𝑝𝜉32 − 𝑔𝑝𝜉12, 𝑔𝑝𝜉33 − 𝜉13)

It is easy to check that this defining ideal is radical. All the other affine charts can be checked

similarly.

53

In Case 3, by [Mae97, Lemma 2.4], the local model has an open affine cover consisting of

three open affine charts. The first two affine charts are both hypersurfaces in A3

𝑅𝑃 and since each is

defined by an irreducible polynomial, each affine chart is reduced. For the third chart, we need to

be careful since it is not a hypersurface. Its coordinate ring is given by

𝑅𝑝 [𝑥, 𝑦, 𝑧, 𝑤]/(𝐹1, 𝐹2),

𝐹1 = 𝑦3 − 𝜔𝑥2𝑤 + (1 − 𝜔)𝑥𝑦𝑧 − 𝑓𝑝,

𝐹2 = 𝑧3 − 𝜔2𝑥𝑤2 − (1 − 𝜔)𝑥𝑦𝑧 − 𝑔𝑝.

for some 𝑓𝑝.𝑔𝑝 ∈ 𝑅𝑝. Here 𝜔 is a primitive third root of unity. Recall that 𝑅𝑝 = 𝒪P3

C[ ¯𝑥1, ¯𝑥2, ¯𝑥3] 𝑆0, where ¯𝑥𝑖 is a coordinate for some standard affine chart in P3
plicative set C[ ¯𝑥1, ¯𝑥2, ¯𝑥3] − 𝑚 𝑝 with 𝑚 𝑝 the maximal ideal associates to 𝑝. Hence:

C,𝑝 =
C, and 𝑆0 is the multi-

𝑅𝑝 [𝑥, 𝑦, 𝑧, 𝑤]/(𝐹1, 𝐹2) (cid:27) (C[ ¯𝑥1, ¯𝑥2, ¯𝑥3, 𝑥, 𝑦, 𝑧, 𝑤]/(𝐹1, 𝐹2))𝑆0

.

We check that this algebra is reduced using Serre’s criterion. Namely, we verify whether our

ring satisfies (𝑅0) and (𝑆1) [sta24b]. Note that 𝐹1, 𝐹2 do not have common factors in the poly-

nomial ring C[ ¯𝑥1, ¯𝑥2, ¯𝑥3, 𝑥, 𝑦, 𝑧, 𝑤]. Hence 𝐹1, 𝐹2 form a regular sequence, and so We have that

C[ ¯𝑥1, ¯𝑥2, ¯𝑥3, 𝑥, 𝑦, 𝑧, 𝑤]/(𝐹1, 𝐹2) is a complete intersection. In particular, this affine chart is Cohen-
Macaulay. This shows C[ ¯𝑥1, ¯𝑥2, ¯𝑥3, 𝑥, 𝑦, 𝑧, 𝑤]/(𝐹1, 𝐹2))𝑆0 is also Cohen-Macaulay and hence sat-
isfies Serre’s condition (𝑆1). On the other hand, one easily checks that 𝐹1 and 𝐹2 intersect

transversally by showing that the rows of the jacobian matrix are never proportional along their

intersection whenever both of them are nonzero. Note that this follows immediately since the

part of the jacobian matrix corresponding to the variables 𝑥, 𝑦, 𝑧, and 𝑤 already satisfies this

property. Hence the 2 × 7 Jacobian matrix of 𝐹1, 𝐹2 is not of full rank if and only if at least

one of the two rows is zero. By the Jacobian criterion, these are precisely the singular points.

We easily see that these points correspond to prime ideals in C[ ¯𝑥1, ¯𝑥2, ¯𝑥3, 𝑥, 𝑦, 𝑧, 𝑤] containing

one of the following ideals:

(𝑥, 𝑦, 𝑓𝑝),(𝑧, 𝑥, 𝑦, 𝑔𝑝),(𝑧, 𝑥, 𝑤, 𝑔𝑝) or (𝑧, 𝑦, 𝑤, 𝑔𝑝). Hence singular

set of C[ ¯𝑥1, ¯𝑥2, ¯𝑥3, 𝑥, 𝑦, 𝑧, 𝑤]/(𝐹1, 𝐹2) has codimension at least 3, which remains true by pass-

ing to the localization with respect to 𝑆0 as 𝑓𝑝, 𝑔𝑝 lives in the maximal ideal of 𝒪P3

C,𝑝. Thus

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𝑅𝑝 [𝑥, 𝑦, 𝑧, 𝑤]/(𝐹1, 𝐹2) is regular in codimension 0, namely (𝑅0). Hence this affine chart is also

reduced. This completes the proof.

□

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