ON ORTHOGONAL LOCAL MODELS OF SHIMURA VARIETIES

By

Ioannis Zachos

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2021

ABSTRACT

ON ORTHOGONAL LOCAL MODELS OF SHIMURA VARIETIES

By

Ioannis Zachos

We study local models that describe the singularities of Shimura varieties of non-PEL

type for orthogonal groups at primes where the level subgroup is given by the stabilizer

of a single lattice. In particular, we use the Pappas-Zhu construction and we give explicit

equations that describe an open subset around the “worst” point of orthogonal local models

given by a single lattice. These equations display the affine chart of the local model as a

hypersurface in a determinantal scheme. Using this we prove that the special fiber of the

local model is reduced and Cohen-Macaulay.

Moreover, by using the explicit description of this affine chart, we resolve the singularities

of our local model. By combining results of Kisin and Pappas, this leads to the construction

of regular p-adic integral models for the corresponding orthogonal Shimura varieties.

ACKNOWLEDGMENTS

I have been studying at Michigan State University for six years and looking back at these

years, is very clear that attaining my Ph.D. is something I could have never accomplished

alone. I want to thank everyone who has supported me in this journey of excitement, self

discovery, hardship, work, kindness and joy.

First and foremost, I want to thank my advisor George Pappas, without whom this

thesis would not exist in this form. Thank you for letting me struggle when I needed to,

and encouraging me when I could not see the light at the end of the tunnel. In all of our

meetings you were full of hope and positivity. Your advice and counsel over the years have

been absolutely indispensable, and your decency and humility are inspiring. Thank you for

being next to me through this whole trip.

I am very grateful to professor Rajesh Kulkarni for running his reading seminar on Stacks

from where I learned so many things. I also want to thank the rest of my committee members,

professors Aaron Levin and Michael Shapiro.

Teaching has become an important part of my life as a mathematician. I want to express

my gratitude to Andy Krause and Tsveta Sendova for their advice, trust, support and mostly

for communicating the joy of teaching to me and everyone in the math department. I want

also to thank Dave Bramer for the nice collaboration that we had in Business Calculus and

Ryan Maccombs for helping me learn how to code in PG for WebWork.

Graduate school is not easy and I think it is impossible if you go through it alone. I

want to thank all my friends for accompanying me in this trip. First, I want to express my

gratitude to the whole “Greek gang”. My roommate and office mate Christos for these years

of friendship. He was always there with his jokes and his vigilant attitude. Mihalis and Ana

iii

Maria with whom I spent numerous nights on my balcony talking about life and existential

matters. Georgios for his interesting discussions on politics and his great skills as a goalie.

Dimitris for his amazing cooking and the game nights that we had together. Andriana for

her infinite patience and her positivity. Ilias and Eleni for bringing Maria to the group and

for all these movie&popcorn nights that we shared.

Escaping to Ann-Arbor was quite common for our group these years mainly due to

Alexandros and Christina who were always ready for our surprise visits. Thank you for all

the good time we had together.

I also want to thank Abhishek and Rodrigo for playing soccer together, Wenchuan for all

the movie nights and Arman for all the tennis and the thought provoking discussions that

we had. I want to thank John and Amalia for their willingness to show us the American

culture and for the numerous trips that we took together. Finally, thank you to all my fellow

participants of the Student Arithmetic Geometry Seminar at MSU, especially to my friends

Armstrong and Nick, who inspired me to work harder.

Because of the pandemic, I spent my last year as graduate student in Thessaloniki, Greece.

I want to thank the math department of Aristotle University in Greece for providing me an

office and I especially want to thank the professors Charalampous and Psaroudakis for their

help.

I want to thank my new/old friends: Pavlos, Nikos, Giannis, Emmeleia, Katerina

and Nikoletta for all the great time that we had together during the lockdown and their

support. I would like to thank my good friends: Neos, Fotis, Vasileia and Euthimia for all

the interesting conversations that we had together. Also, I would like to thank my friends

from Larisa: Ilias, Bakoulas, Thanasis, Kopas, Labrini, Dotas, Dasios and Vaios for all the

talks and the fun time that we had together.

Last, I want to express my gratitude to my family, without whom nothing could have

iv

been made possible. Thank you Mom and Dad for trusting in me, for pushing me, for letting

me go. Sisters, thank you for the nice adventures that we had in the US and for all the great

memories that I have with you.

I thank God for everyone who came in my life and in some direct or indirect way has

helped me complete this dissertation. Thank you!

v

TABLE OF CONTENTS

KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Local models
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Normal forms of quadric lattices . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 An Affine Chart of Mloc(Λ)

. . . . . . . . . . . . . . . . . . . . . .
3.1 Lattices over O[u] and orthogonal local models . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The affine chart Ud,l

Chapter 4 Reduction of Relations of Ud,l

. . . . . . . . . . . . . . . . . . . .
4.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 S/I(cid:48) (cid:39) S(cid:48)(cid:48)/I(cid:48)(cid:48). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I = I(cid:48).

Chapter 5 Flatness of Ud,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10
10
12
13

16
16
19

25
26
30

35

Chapter 6 Reducedness of U d,l . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Chapter 7

Irreducible Components of U d,l Part I . . . . . . . . . . . . . . .
l = 2 and l = d − 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 < l < d − 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1
7.2

Chapter 8 The Remaining Cases

. . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Proof of Theorem 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 9

Irreducible Components of U d,l Part II . . . . . . . . . . . . . . .
(d, l) = (odd, odd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(d, l) = (odd, even)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(d, l) = (even, odd)

9.1
9.2
9.3

Chapter 10 The Blow-Up of Mloc(Λ) . . . . . . . . . . . . . . . . . . . . . . . .

40
40
45

50
50

52
52
53
54

55

Chapter 11 Application to Shimura Varieties . . . . . . . . . . . . . . . . . .

59

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

vi

KEY TO SYMBOLS

an odd prime
a finite field extension of Qp

the ring of integers of F
a uniformizer of OF

the residue field of F

an algebraic closure of F

the completion of the maximal unramified extension of F in ¯F

the ring of integers of ˘F

the residue field of ˘F

the dimension of the F -vector space V
an OF -lattice in V

the dual of Λ in V
the distance of the lattice Λ to its dual Λ∨
i=1O[u] · ¯ei
the O[u]-lattice given by L = ⊕d
the polynomial ring over O with variables the entries of the matrix (B1|B2)

the leading term of the polynomial f
the 2 × 2 minors of the matrix (B1|B2)

the Pappas-Zhu local model

an affine chart of Mloc(Λ) around the worst point

the special fiber of Ud,l

p

F
OF

π

κF

¯F

˘F
O

k

d

Λ
Λ∨

l

L

O[B1|B2]

LT (f )

∧2(B1(cid:124)B2)

Mloc(Λ)

Ud,l

U d,l

vii

Chapter 1

Introduction

Local models of Shimura varieties are projective flat schemes over the spectrum of a discrete

valuation ring. These projective schemes are expected to model the singularities of integral

models of Shimura varieties with parahoric level structure. The definition of local model

was formalized to some degree by Rapoport and Zink in [20]. However, it was soon realized

that the Rapoport-Zink construction is not adequate when the group of the Shimura variety

is ramified at p and in many cases of orthogonal groups. Indeed, then the corresponding

integral models of Shimura varieties are often not flat ([14]).

In [19], Pappas and Zhu

gave a general group theoretic definition of local models. These local models appear as

subschemes of global (“Beilinson-Drinfeld”) affine Grassmannians and are associated to local
model triples. A LM-triple over a finite extension F of Qp, for p (cid:54)= 2, is a triple (G,{µ}, K)
consisting of a reductive group G over F , a conjugacy class of cocharacters {µ} of G over an
K (G,{µ})

algebraic closure of F , and a parahoric subgroup K of G(F ). We denote by Mloc

the corresponding local model.

In the present thesis, we study local models for Shimura varieties for forms of the orthog-

onal group which are of Hodge but not PEL type. An example of such a Shimura variety is

the following: Consider the group G = GSpin(V), where V is a (non-degenerate) orthogonal
space of dimension d ≥ 7 over Q and the signature of VR is (d− 2, 2). Let D be the space of

oriented negative definite planes in VR. Then the pair (G, D) is a Shimura datum of Hodge

1

type. Further, consider a Zp-lattice Λ in V = V ⊗Q Qp, for which

pΛ∨ ⊂ Λ ⊂ Λ∨,

where Λ∨ is the dual of Λ for the corresponding symmetric form. We denote by l the distance
of the lattice Λ to its dual Λ∨, i.e. l = lgZp(Λ∨/Λ) and we set l∗ = min(l, d − l). We let K1
be the connected stabilizer of Λ in SO(V)(Qp) and let K be the corresponding parahoric
subgroup of G(Qp). The group G is the smooth connected “Bruhat-Tits” group scheme
over Spec (Zp) such that G ⊗Zp
Qp = G ⊗Q Qp and G(Zp) = K. Now, for a compact open
subgroup K ⊂ G(Af ) of the form K = Kp · Kp where Kp = K and Kp is sufficiently small,

the corresponding Shimura variety is

ShK(G, D) = G(Q)\(D × G(Af )/K).

This complex space has a canonical structure of an algebraic variety over the reflex field Q

(see [13]). The work of Kisin and Pappas [10] gives that orthogonal Shimura varieties as

above admit integral models SK(G, D), whose singularities are the “same” as those of the

corresponding PZ local models; see Theorem 1.0.1 below where the properties (a) and (b)

imply that SK(G, D) and the corresponding local model are locally isomorphic for the ´etale

topology. Note that there is a central extension (see [11])

1 → Gm → GSpin(V ) → SO(V ) → 1.

Hence, by [8, Proposition 2.14], the local model that pertains to the above Shimura va-
(SO(V ),{µ}) for the LM triple (SO(V ),{µ}, K1) where V , K1,

riety is Mloc(Λ) = Mloc
K1

2

are as above and we take the minuscule coweight µ : Gm → SO(V ) to be given by
µ(t) = diag(t−1, 1, . . . , 1, t). In fact, we will consider a more general situation in which
Qp is replaced by a finite field extension F of Qp with integers OF . As a special case of [10,

Theorem 4.2.7] we have the following:

Theorem 1.0.1. There is a scheme SK(G, D), flat over Spec (Zp), with

SK(G, D) ⊗Zp

Qp = ShK(G, D) ⊗Q Qp,

and which supports a “local model diagram”

(cid:102)SK(G, D)

πK

SK(G, D)

such that:

qK

Mloc(Λ)

(1.0.0.1)

a) πK is a G-torsor for the parahoric group scheme G that corresponds to Kp,

b) qK is smooth and G-equivariant.

Let us add that the integral model SK(G, D) satisfies several additional properties, see
[10] and [18, §7]. It is also “canonical” in the sense of [16]. At this point, we want to mention

that one application of such orthogonal Shimura varieties lies in arithmetic intersection

theory. For example, orthogonal Shimura varieties are used in the proof of the averaged

Colmez conjecture (see [3] and [2]).

In the rest of the thesis, we will mainly consider local models and Shimura varieties will

only appear again in Chapter 11 where we discuss how our results apply to GSpin Shimura

3

varieties. Moreover, we want to mention that the results in Chapters 10 and 11 are from the

joint work [18].

In this thesis, we first give an explicit description of Mloc(Λ). The difficulty in this

task arises from the fact that the construction of PZ local models is inexplicit and group

theoretical. In particular, in order to define the PZ local model we have to take the reduced

Zariski closure of a certain orbit inside a global affine Grassmannian. We refer the reader to

Section 2.1 where the construction of the PZ local models is reviewed. In the case of local

models of PEL type one can use the standard representation of the group to quickly represent

the local model as a closed subscheme of certain linked (classical) Grassmannians (see [19]).
This is not possible here since the composition i · µ, where i : SO(V ) (cid:44)→ GL(V ) is the

natural embedding, is not a minuscule coweight and we have to work harder. Nevertheless,

we give explicit equations for an affine chart of the “worst” point of the local model. These

equations display this chart as a quadric hypersurface given by the vanishing of a trace in
a determinantal scheme of 2 × 2 minors. Using this and classical results on determinantal

varieties we prove that the special fiber of the affine chart is reduced and Cohen-Macaulay.

This implies that the special fiber of the local model is reduced and Cohen-Macaulay. Note

here that the “reduced” result follows from Pappas-Zhu paper [19], which in turn uses Zhu’s

proof of the Pappas-Rapoport coherence conjecture (see [22]). We want also to mention the

recent work of Haines and Richarz [7], where the authors prove in a more general setting

that the special fiber of the PZ local models is reduced and Cohen-Macaulay.

Here, we give an independent elementary proof of these properties by using the explicit

equations which, as we said above, describe an open subset around the “worst” point of our

local model. We also calculate the number of the irreducible components of the special fiber

of the affine chart. This is equal to the number of irreducible components of the special fiber

4

of the local model. The reason behind these implications lies in the construction of the local

model. In particular, as discussed in [19] the geometric special fiber of the PZ local model

is a union of affine Schubert varieties. Among those there is a unique closed orbit which

consists of a single point, the “worst” point. The one-point stratum lies in the closure of

every other stratum. It follows that, if the special fiber of the local model has a certain nice

property at the worst point (for example reducedness), then this should hold everywhere

(see for example [6]).

Moreover, from the above discussion and the construction of the local models in [19], we

deduce that our affine chart is dense and hence it “captures all the singularities” of Mloc(Λ).

By using the explicit description of this affine chart, we prove that the blow up of Mloc(Λ)

at the worst point resolve the singularities (see Theorem 1.0.3), which in turn leads to the

construction of regular integral models for the Shimura varieties ShK(G, D) over the p-adic
integers Zp. We expect that this construction will find applications to the study of arithmetic

intersections of special cycles and Kudla’s program.

Below we denote by O the ring of integers of ˘F , which is the completion of the maximal

unramified extension of F in a fixed algebraic closure, and by k the residue field of ˘F .

The thesis is organized as follows:

In Chapter 2 we review the definition of the PZ

local models. In Chapter 3 we show how we derive the explicit equations. We describe an
affine chart of the worst point ∗ of our orthogonal local model in the cases where (d, l) =

(even, even), (d, l) = (odd, odd), (d, l) = (even, odd) and (d, l) = (odd, even). Note that
when l is even the symmetric form on V ⊗F
on V ⊗F

˘F splits and when l is odd the symmetric form

˘F is quasi-split but not split.

The case that l∗ ≤ 1, has been considered by Madapusi Pera in [11] and also in the joint

work of He, Pappas and Rapoport [8]. In the last chapter of [8], the authors easily prove

5

that in the case l∗ = 0 the local model is isomorphic to a smooth quadric. With some more

work they prove that in the case l∗ = 1 the local model is isomorphic to a quadric which
is singular in one point. Here, we assume that l∗ > 1 (and also d ≥ 5) and extend these

results.

Before stating our main theorems we need some more notation. Thus, let n = (cid:98)d/2(cid:99),

r = (cid:98)l/2(cid:99) and X be a d × d matrix of the form:



X =

E1 O1 E2

B1 A B2

E3 O2 E4

 ,

where Ei ∈ Mat(n−r)×(n−r), Oj ∈ Mat(n−r)×l, B(cid:96) ∈ Matl×(n−r) and A ∈ Matl×l. We write
O[X], O[B1|B2] for the polynomial rings over O with variables the entries of the matrices
X and (B1|B2) respectively. We also write ∧2(B1(cid:124)B2) for the 2 × 2 minors of (B1|B2) and

Jm for the unit antidiagonal matrix of size m,



Jm :=

. . .

1

1

 .

In the introduction, we state our results in the case that d and l have the same parity, so

d = 2n and l = 2r, or d = 2n + 1 and l = 2r + 1. The results when d and l have different

parity are a bit more involved to state; we refer the reader to Theorem 3.2.2 and Chapter 8.

Theorem 1.0.2. Suppose that d and l have the same parity. Then an affine chart of the
local model Mloc(Λ) around the worst point ∗ is given by Ud,l = Spec (R) where R is the

6

quotient ring

R = O[B1|B2]/(∧2(B1(cid:124)B2), T r(B2Jn−rBt

1Jl) + 2π).

Let us mention here that, for l∗ (cid:54)= 0, none of these models are smooth or semi-stable (as

follows from [8, Theorems 5.1, 5.6]).

In Chapters 10 and 11, we resolve the singularities of Mloc(Λ) and SK(G, D) respectively.
We consider the blow-up of Mloc(Λ) at the point ∗. This gives a G-birational projective

morphism

rbl : Mbl(Λ) −→ Mloc(Λ).

Using the explicit description of Ud,l above, we show:

Theorem 1.0.3. The scheme Mbl(Λ) is regular and has special fiber a divisor with normal
crossings. In fact, Mbl(Λ) is covered by open subschemes which are smooth over Spec (Zp[u, x,
y]/(u2xy − p)).

We see that the corresponding blow-up S reg

K (G, D) of the integral model SK(G, D)

inherits the same nice properties as Mbl(Λ).

In fact, there is a local model diagram for

S reg

K (G, D) similar to (1.0.0.1) but with Mloc(Λ) replaced by Mbl(Λ). See Theorem 11.0.1

for the precise statement about the model S reg

integral models for ShK(G, D). The construction of S reg

K (G, D); this theorem gives regular p-adic
K (G, D) from rbl and the local

model diagram (11.0.0.1) is an example of a “linear modification” in the sense of [14].

Below we discuss how we derive the equations of Theorem 1.0.2 and then we give the

main ingredients of the proof.

We write S0, S1 for the antidiagonal matrices of size d,

7



S0 :=

0(l)

1(n−r)

1(n−r)

 , S1 :=



1(l)

0(n−r)

0(n−r)



and we define the ideal

(cid:16)

Inaive =

X2, ∧2X, XtS0X − 2π (S0 + πS1) X, XtS1X + 2(S0 + πS1)X

(cid:17)

.

Our first step is to show that an affine chart of the PZ local model around the worst point ∗
is given as a closed subscheme of the quotient M = O[X]/Inaive. We do this in Chapter 3.
This O-flat closed subscheme is obtained by adding certain equations to Inaive: Set

I = Inaive + Iadd

where

(cid:16)

Iadd =

T r(X), T r(A) + 2π, B2Jn−rBt

1 − AJl

(cid:17)

.

We show that I cuts out the O-flat Mloc(Λ) ∩ M, which is an open affine subscheme of

Mloc(Λ). By an involved but completely elementary manipulation of the relations describing

the ideal I we prove that:

Theorem 1.0.4. Suppose that d and l have the same parity. The quotient O[X]/I is iso-

morphic to O[B1|B2]/(cid:0)∧2(B1(cid:124)B2), T r(B2Jn−rBt

1Jl) + 2π(cid:1) .

It essentially remains to show that Ud,l = Spec (R) is flat over O. By definition, Ud,l
is a hypersurface in the determinantal scheme D = Spec (O[B1|B2] /(∧2(B1|B2)). Since

D is Cohen-Macaulay, see [21, Remark 2.12], we can easily deduce that Ud,l and U d,l are

8

also Cohen-Macaulay. Flatness of Ud,l follows, see Chapter 5. Theorem 1.0.2 quickly follows

together with the (essentially equivalent) statement:

Theorem 1.0.5. Suppose that d and l have the same parity. An affine chart of the local
model Mloc(Λ) around the worst point is given by Spec (O[X]/I), where I is as above.

Using Theorem 1.0.2 and the reducedness of the fibers of PZ local models (see [19]) we

have that:

Theorem 1.0.6. The special fiber of Ud,l is reduced.

In Chapter 6 we give an independent proof of this result by using that the special fiber

U d,l is Cohen Macaulay and generically reduced.

In the course of proving the reducedness of U d,l, we also determine the number of its
irreducible components. We find that when 2 < l∗, where l∗ = min(l, d − l) and l is the

distance of our lattice to its dual, the special fiber U d,l has two irreducible components.

When l∗ = 2, U d,l has three irreducible components.

In fact, we explicitly describe the

equations defining the irreducible components of the special fiber. Similar arguments extend

to the case that d and l have different parity. We give the corresponding hypersurface in a

determinantal scheme and the equations of irreducible components of the special fiber in all

cases.

9

Chapter 2

Preliminaries

Let us fix an odd prime p and consider a finite field extension F/Qp. Denote with OF the
ring of integers of F and let π be a uniformizer of OF . We denote by ˘F the completion
of the maximal unramified extension of F in an algebraic closure ¯F . We denote by κF the

residue field of F and by k the algebraic closure of κF which is also the residue field of ˘F .
We also set O := O ˘F for the ring of integers of ˘F .

2.1 Local models

We now recall the construction of the Pappas-Zhu local models. For a more detailed presen-

tation we refer the reader to [15] and [19].

Let G be a connected reductive group over F . Assume that G splits over a tamely ramified

extension of F . Let {µ} be a conjugacy class of a geometric cocharacter µ : G

m ¯F → G ¯F

and assume that µ is minuscule. Define K to be the parahoric subgroup of G(F ), which is
the connected stabilizer of some point x in the (extended) Bruhat-Tits building B(G, F ) of

G(F ). Define E to be the extension of F which is the field of definition of the conjugacy
class {µ} (the reflex field).

In [19], the authors construct an affine group scheme G which is smooth over Spec (OF [t])

and which, among other properties, satisfies:

10

1. The base change of G by Spec (OF ) → Spec (OF [t]) = A1
OF

given by t → π is the

Bruhat-Tits group scheme which corresponds to the parahoric subgroup K (see [1]).

2. The group scheme G|OF [t,t−1] is reductive.

Next, they consider the global (“Beilinson-Drinfeld”) affine Grassmannian

AffG,A1

OF

→ A1
OF

given by G, which is an ind-projective ind-scheme. By base changing t → π, they obtain an

equivariant isomorphism

AffG

∼−→ AffG,A1

OF

×A1
OF

Spec (F )

where AffG is the affine Grassmannian of G; this is the ind-projective ind-scheme over

Spec (F ) that represents the fpqc sheaf associated to

R → G(R((t)))/G(R[[t]]),

where R is an F -algebra (see also [17]).

The cocharacter µ gives an ¯F [t, t−1]-valued point of G and thus µ gives an ¯F ((t))-valued

point µ(t) of G. This gives a ¯F -point [µ(t)] = µ(t)G( ¯F [[t]]) of AffG. Since µ is minuscule
and {µ} is defined over the reflex field E the orbit

G( ¯F [[t]])[µ(t)] ⊂ AffG( ¯F ),

is equal to the set of ¯F -points of a closed subvariety Xµ of AffG,E = AffG ⊗F E.

11

Definition 2.1.0.1. Define the local model Mloc

K (G,{µ}) to be the flat projective scheme

over Spec (OE) given by the reduced Zariski closure of the image of

Xµ ⊂ AffG

∼−→ AffG,A1

OF

×A1
OF

Spec (E)

in the ind-scheme AffG,A1

OF

×A1
OF

Spec (OE).

The PZ local models have the following property (see [8, Prop. 2.14]).

Proposition 2.1.0.2. If F(cid:48)/F is a finite unramified extension, then

K (G,{µ}) ⊗OE
Mloc

OE(cid:48) ∼−→ Mloc

K(cid:48)(G ⊗F F(cid:48),{µ ⊗F F(cid:48)}).

Note that here the reflex field E(cid:48) of (G ⊗F F(cid:48),{µ ⊗F F(cid:48)}) is the join of E and F(cid:48). Also, K(cid:48)
is the parahoric subgroup of G ⊗F F(cid:48) with K = K(cid:48) ∩ G.

The above proposition allows us to base change to an unramified extension F(cid:48) over F .

This will play a crucial role in the proof of our main theorems.

2.2 Quadratic forms

Let V be an F -vector space with dimension d = 2n or 2n+1 equipped with a non-degenerate
symmetric F -bilinear form (cid:104) , (cid:105). It follows from the classification of quadratic forms over
local fields [5] that after passing to a sufficiently big unramified extension F(cid:48) of F , the base
change of (V,(cid:104) , (cid:105)) to F(cid:48) affords a basis as in one of the following cases:

1. Split form: there is a basis fi with the following relations:

(cid:104)fi, fd+1−j(cid:105) = δij, ∀i, j ∈ {1, . . . , d}.

12

2. Quasi-split form (for d = 2n): there is a basis fi with the relations: (cid:104)fi, fd+1−j(cid:105) =

δij, for i, j (cid:54)= n, n + 1, (cid:104)fn, fn(cid:105) = π, (cid:104)fn+1, fn+1(cid:105) = 1, (cid:104)fn, fn+1(cid:105) = 0.

3. Quasi-split form (for d = 2n + 1): there is a basis fi with the relations: (cid:104)fi, fd+1−j(cid:105) =

δij, for i, j (cid:54)= n + 1, (cid:104)fn+1, fn+1(cid:105) = π.

2.3 Normal forms of quadric lattices

Let V be an F -vector space with dimension d = 2n or 2n+1 equipped with a non-degenerate
symmetric F -bilinear form (cid:104) , (cid:105). We assume that d ≥ 5. For all the cases below we take the
minuscule coweight µ : Gm → SO(V ) to be given by µ(t) = diag(t−1, 1, . . . , 1, t), defined

over F .

A lattice Λ ⊂ V is called a vertex lattice if Λ ⊂ Λ∨ ⊂ π−1Λ. By Λ∨ we denote the dual

of Λ in V :

Λ∨ := {x ∈ V |(cid:104)Λ, x(cid:105) ⊂ OF}.

Let Λ in V be a vertex lattice. So, Λ ⊂l Λ∨ ⊂l(cid:48) π−1Λ with l + l(cid:48) = d. Here l (respectively
l(cid:48)) is the length l = lg(Λ∨/Λ) (respectively l(cid:48) = lg(π−1Λ/Λ∨)). We assume that l > 1 and
l(cid:48) > 1.

For the following we refer the reader to Rapoport-Zink’s book [20], Appendix on Normal

forms of lattice chains. More precisely, by [20, Appendix, Proposition A.21], after an ´etale
base change (i.e an unramified base change) we can find an OF -basis {ei} of Λ with the

following property:

For d = 2n:

13

1. Split form: Λ = ⊕d

i=1OF · ei with

(cid:104)ei, ed+1−j(cid:105)= δij, for i (cid:54)∈ [n − r + 1, n + r],

(cid:104)ei, ed+1−j(cid:105)= πδij, for i ∈ [n − r + 1, n + r].

We have Λ⊂l Λ∨ where l = 2r.

2. Quasi-split form: Λ = ⊕d

i=1OF · ei with

(cid:104)ei, ed+1−j(cid:105)= δij, for i ∈ [1, d] \ [n − r, n + r + 1],

(cid:104)ei, ed+1−j(cid:105)= πδij, for i ∈ [n − r, n + r + 1] \ {n, n + 1},

(cid:104)en, en(cid:105) = π,

(cid:104)en+1, en+1(cid:105) = 1,

(cid:104)en, en+1(cid:105) = 0.

We have Λ⊂l Λ∨ where l = 2r + 1.

For d = 2n + 1:

3. Split form: Λ = ⊕d

i=1OF · ei with

(cid:104)ei, ed+1−j(cid:105)= δij, for i (cid:54)∈ [n + 1 − r, n + 1 + r] \ {n + 1},

(cid:104)ei, ed+1−j(cid:105)= πδij, for i ∈ [n + 1 − r, n + 1 + r] \ {n + 1}.

We have Λ⊂l Λ∨ where l = 2r.

14

4. Quasi-split form: Λ = ⊕d

i=1OF · ei with

(cid:104)ei, ed+1−j(cid:105)= δij, for i (cid:54)∈ [n + 1 − r, n + 1 + r],

(cid:104)ei, ed+1−j(cid:105)= πδij, for i ∈ [n + 1 − r, n + 1 + r].

We have Λ⊂l Λ∨ where l = 2r + 1.

From the above discussion, it follows that we can reduce our problem to the above cases

by passing to a sufficiently big unramified extension of F . Thus, from now on we will be
working over ˘F . Recall that we denote by O its ring of integers and by k its residue field.
In all cases, we will denote by S the (symmetric) matrix with entries (cid:104)ei, ej(cid:105) where {ei}

is the basis above. We can then write

S = S0 + πS1

where S0, S1 both have entries only 0 or 1. For example, in case (1) we have the anti-diagonal

matrices:



S0 :=

0(2r)

1(n−r)

1(n−r)

 , S1 :=



1(2r)

0(n−r)

0(n−r)

 .

15

Chapter 3

An Affine Chart of Mloc(Λ)

3.1 Lattices over O[u] and orthogonal local models

We can now extend our data to O[u, u−1]. We define V = ⊕d
V× V → O[u, u−1] a symmetric O[u, u−1]-bilinear form such that the value of (cid:104)¯ei, ¯ej(cid:105) is the

i=1O[u, u−1]¯ei and (cid:104) , (cid:105)

:

same as the above for V with the difference that π is replaced by u. Similarly, we define
¯µ(t) : Gm → SO(V) by using the {¯ei} basis for V.
We also define L the O[u]-lattice in V by L = ⊕d

i=1O[u] · ¯ei. From the above we see that

the base change of (V, L,(cid:104) , (cid:105)) from O[u, u−1] to F given by u (cid:55)→ π is (V, Λ,(cid:104) , (cid:105)).

Let us now define the local model Mloc(Λ) = Mloc

K (SO(V ),{µ}) where K is the parahoric
stabilizer of Λ. We consider the smooth, as in [19], affine group scheme G over O[u] given
by g ∈ SO(V) that also preserves L and L∨. If we base change by u (cid:55)→ π we obtain the
Bruhat-Tits group scheme G of SO(V ) which is the stabilizer of the lattice chain Λ ⊂ Λ∨ ⊂
π−1Λ. The corresponding parahoric group scheme is the neutral component G0 of G. The
construction of [19] produces the group scheme G0 that extends G0. By construction, there
is a group scheme immersion G0 (cid:44)→ G.

In this case, the global (“Beilinson-Drinfeld”) affine Grassmannian

→ Spec (O[u])

AffG,A1O

16

represents the functor that sends the O[u]-algebra R, given by u (cid:55)→ r, to the set of projective
finitely generated R[u]-modules L of V ⊗O R which are locally free such that (u − r)NLR ⊂
L ⊂ (u − r)−NLR for some N >> 0 and satisfy

L⊂ l L∨ ⊂ l(cid:48) u−1L

with all graded quotients R-locally free and of the indicated rank. Here, we set LR = L⊗O R.
Consider the O-valued point [L(0)] given by L(0) = ¯µ(u−π)L. Then, as in the Section 2.1
×A1O Spec (O);
the local model is the reduced Zariski closure of the orbit [L(0)] in AffG0,A1O
it inherits an action of the group scheme G0 = G0 ⊗O[u] O. As in [19], there is a natural
→ AffG,A1O induced by G0 (cid:44)→ G which identifies Mloc(Λ) with a closed
×A1O Spec (O) .
By the definition of L(0) we have

morphism AffG0,A1O
subscheme of AffG,A1O

(u − π)L ⊂ L(0) ∩ L

⊂ L
⊂
L(0)

⊂

⊂

L + L(0) ⊂ (u − π)−1L,

where the quotients along all slanted inclusions are O-free of rank 1 (for more details see
×A1O Spec (O)

proof of Proposition 3.1.0.1). Let us define M to be the subfunctor of AffG,A1O
that parametrizes all L such that

(u − π)L ⊂ L ⊂ (u − π)−1L.

Then M is represented by a closed subscheme of AffG,A1O
In that way, Mloc(Λ) is a closed subscheme of M and Mloc(Λ) is the reduced Zariski closure of

×A1O Spec (O) which contains [L(0)].

17

its generic fiber in M. As in [8, Proposition 12.7], the elements of Mloc(Λ) have the following

properties:

Proposition 3.1.0.1. If L ∈ Mloc(Λ)(R), for an O-algebra R, then:

1. L is u-stable,

2. L⊂ l L∨, and

3.

LR
(u − π)LR ⊂ L ∩ LR⊂
L⊂

⊂

⊂

LR + L ⊂ (u − π)−1LR,

where the quotients arising from all slanted inclusions are generated as R-modules by
one element (we say that they have rank ≤ 1).

Proof. The first two conditions follow directly from the definition of the local model. By the
definition of L(0) we have L(0) = ¯µ(u−π)L where ¯µ(u−π) = diag((u−π)−1, 1, . . . , 1, u−π).
We can easily see that (3) is true for L(0). Since condition (3) is closed and G-equivariant it
also holds for L and the proposition follows.

Define F(cid:48) to be the image of L by the map

(u − π)−1LR/(u − π)LR

u−π−−−→ LR/(u − π)2LR.

Define the symmetric bilinear form:

(cid:104) , (cid:105)(cid:48) : L/(u − π)2L × L/(u − π)2L → O[u]/(u − π)2O[u],

18

by

(cid:104) , (cid:105)(cid:48) = (cid:104) , (cid:105)mod(u − π)2.

Notice, that condition (2) above means that (cid:104)L,L(cid:105) ∈ R[u] under the R-base change of the
bilinear form (cid:104) , (cid:105). Thus, F(cid:48) is isotropic for (cid:104) , (cid:105)(cid:48) on LR/(u − π)2LR × LR/(u − π)2LR, i.e.
(cid:104)F(cid:48),F(cid:48)(cid:105)(cid:48) = 0. We also observe that rank(u − π) ≤ 1 where u − π : F(cid:48) → F(cid:48). That follows
from condition (3) and the fact that (u − π)2LR = 0 in LR/(u − π)2LR.

3.2 The affine chart Ud,l

For the sake of simplicity we fix d = 2n and l = 2r. We get similar results for all the other

cases.

For any O-algebra R, let us consider the R-submodule:

F = {(u − π)v + Xv | v ∈ Rd} ⊂ (u − π)Rd ⊕ Rd ∼= LR/(u − π)2LR

with X ∈ Matd×d(R).

We ask that F satisfies the following three conditions:

1. u-stable:

It suffices to be (u − π)-stable. Let (u − π)v + Xv ∈ F. Then there
exists w ∈ Rd such that (u − π)2v + (u − π)Xv = (u − π)w + Xw. This gives
Xuv − Xπv = uw − πw + Xw and so:

w = Xv,

−πXv = −πw + Xw.

19

By substituting the former equation to the latter, we have X2v = 0. Because this is

correct for every v, we have X2 = 0. Observe that X is the matrix giving multiplication
by (u − π) on F.

2. Isotropic: Let (u − π)v + Xv ∈ F. We want

(cid:104)(u − π)v + Xv, (u − π)v + Xv(cid:105)(cid:48) = 0

and recall that (cid:104) , (cid:105)(cid:48) = (cid:104) , (cid:105)mod(u − π)2. By simplifying the above equation we have

−2(u − π)(cid:104)v, Xv(cid:105)(cid:48) = (cid:104)Xv, Xv(cid:105)(cid:48).

The above relation holds for any v and so we get:

−2(u − π)(S0 + uS1)X = Xt(S0 + uS1)X

where S0, S1 are the matrices with S = S0 + πS1 = ((cid:104)ei, ej(cid:105))i,j as in Section 2.3. By

simplifying the above relation we have:

2πS0X + 2π2S1X + u(−2πS1X − 2S0X) = XtS0X + u(XtS1X)

which amounts to

XtS0X = 2π(S0X + πS1X) and XtS1X = −2(S0X + πS1X).

3. rank(u−π| F(cid:48) )≤1: By the above, this translates to ∧2X = 0.

20

Let U naive be the corresponding scheme of F defined by the d × d matrices X which

satisfy the following relations:

X2 = 0, XtS0X − 2π(S0X + πS1X) = 0,

∧2X = 0, XtS1X + 2(S0X + πS1X) = 0.

We denote by Inaive the ideal generated by the entries of the above relations.

The conditions (1)-(3) are necessary but not always sufficient for L to correspond to
an R-valued point of Mloc(Λ). Indeed, the generic fiber of U naive contains the additional
˘F -point L = L ˘F which is not in the orbit [L(0)] of ¯µ in the affine Grassmannian AffG. Also,
calculations in low dimensions show that U naive has non-reduced special fiber.

Our goal is to calculate the O-flat closed subscheme U = Mloc(Λ) ∩ U naive of U naive by
adding some explicit relations in the ideal Inaive. The resulting U is an open subscheme of

Mloc(Λ).

Observe that the point L is fixed by the action of the group scheme G0 and so its our worst
point. Thus, U is an open neighborhood around the worst point L. Then these additional

relations, together with Inaive, give explicit equations that describe an open subset around

the worst point of our local model Mloc(Λ).

We introduce some notation that will help us defining those relations. We first rewrite

our matrix X := (xij)1≤i,j≤d as follows:

X =





E1 O1 E2

B1 A B2

E3 O2 E4

21

where Ei ∈ Mat(n−r)×(n−r), Oj ∈ Mat(n−r)×l, B(cid:96) ∈ Matl×(n−r) and A ∈ Matl×l. We
denote by O[X] the polynomial ring over O, with variables the entries of the matrix X. We

also write Jm for the unit antidiagonal matrix of size m,



1

 .

Jm :=

. . .

1

We will show that by adding the following relations:

T r(X) = 0, T r(A) + 2π = 0, B2Jn−rBt

1 − AJl = 0,

we get the desired O-flat scheme U in the cases where (d, l) =(even,even) and (d, l) =

(odd,odd). By adding similar relations we get the corresponding result in cases where

(d, l) =(even,odd) and (d, l) =(odd,even). Next, we state the main theorems of this the-

sis.

Theorem 3.2.1. Suppose that d and l have the same parity so d = 2n and l = 2r, or

d = 2n + 1 and l = 2r + 1. Then an affine chart of the local model Mloc(Λ) around the worst
point is given by Ud,l = Spec (O[X]/I), which is defined by the quotient of the polynomial
ring O[X] = O[(xi,j)1≤i,j≤d] by the ideal

I = Inaive + Iadd

where

(cid:16)

Iadd =

T r(X), T r(A) + 2π, B2Jn−rBt

1 − AJl

(cid:17)

.

22

Next, we state the theorems for the cases where d and l have different parity. In each
case we consider d× d matrices X. In order to define the submatrices (Ei, Oj, B(cid:96), A) giving

the block decomposition of X we set:

 r

r(cid:48) =

if l = 2r

r + 1 if l = 2r + 1.

Then write the matrix X as before, with blocks Ei∈ Mat(n−r(cid:48))×(n−r(cid:48)), Oj ∈ Mat(n−r(cid:48))×(l+1),
A∈ Mat(l+1)×(l+1) and B(cid:96)∈ Mat(l+1)×(n−r(cid:48)).

in the (n + 1)-row and (n + 1)-column of X. Similarly we denote by B(cid:48)

We denote by A(cid:48) the l × l matrix which is obtained from A by erasing the part that is
2 the l × (n − r(cid:48))
matrices which are obtained from B1, B2 by erasing the part that is on the (n + 1)-row of
X. Lastly, we denote by Q the (r(cid:48) + 1)-column of A and Q(cid:48) the (r(cid:48) + 1)-column of A with

1, B(cid:48)

the (n + 1)-entry erased.

Theorem 3.2.2. Suppose that d and l have opposite parity, so d = 2n + 1 and l = 2r or

d = 2n and l = 2r + 1. An affine chart of the local model Mloc(Λ) around the worst point
L = L is given by Ud,l = Spec (O[X]/I), which is defined by the quotient of the polynomial
ring O[X] by the ideal

I = Inaive + Iadd

where

Iadd = (T r(X), T r(A(cid:48)) + 2π, B(cid:48)

2Jn−r(cid:48)(B(cid:48)

1)t +

Q(cid:48)(Q(cid:48))t − A(cid:48)Jl).

1
2

In chapters 4-6 we carry out the proof of Theorem 3.2.1 for the case (d, l) = (even, even).

The proof of the remaining cases of parity for d and l is given in Chapter 8.

Using Theorems 3.2.1, 3.2.2 and the fact that PZ local models have reduced special fiber,

23

see [19], we obtain:

Theorem 3.2.3. The special fiber of Ud,l is reduced.

Note that in the above theorem we do not specify the parity of d and l. In Chapter 6

we give an independent proof of this theorem, for the case (d, l) = (even, even), by showing

that the special fiber is Cohen Macaulay and generically reduced. A similar argument works

for the rest of the cases of parity for d and l.

24

Chapter 4

Reduction of Relations of Ud,l

In all of Chapter 4, we assume d = 2n and l = 2r. Our goal in this chapter is to prove the

simplification of equations given by Theorem 4.0.1. (This corresponds to Theorem 1.0.4 of

the introduction.)

We are working over the polynomial ring S := O[(xi,j)1≤i,j≤d]. We also set

S(cid:48)(cid:48) := O[(xt,s)t∈Z,s∈Zc]

where Z := {n − (r − 1), . . . , n, n + 1, . . . , d − n + r} and Zc := {1, 2, 3, . . . , d} \ Z.

Recall that

I = (X2, ∧2X, T r(X), T r(A) + 2π, B2Jn−rBt

1 − AJ2r,

XtS0X − 2π(S0X + πS1X), XtS1X + 2(S0X + πS1X)).

We set

where

I(cid:48)(cid:48) =

(cid:17)

(cid:16)∧2(B1(cid:124)B2), T r(B2Jn−rBt
∧2(B1(cid:124)B2) := (xi,jxt,s − xi,sxt,j)i,t∈Z, j,s∈Zc.

1J2r) + 2π

25

Theorem 4.0.1. There is an O-algebra isomorphism S/I ∼= S(cid:48)(cid:48)/I(cid:48)(cid:48).

Proof. We define the ideal:

I(cid:48) =

1 − AJ2r,

(cid:16) ∧2 X, T r(X), T r(A) + 2π, B2Jn−rBt
(cid:17)

XtS1X + 2(S0X + πS1X)

.

The proof will be done in two steps:

1. Show I = I(cid:48).

2. Show S/I(cid:48) (cid:39) S(cid:48)(cid:48)/I(cid:48)(cid:48).

4.1

I = I(cid:48).

Our first reduction is to prove that I(cid:48) = I, which will be given in Proposition 4.1.0.8. To
do that, we are going to show that the entries of X2, XtS0X − 2π(S0X + πS1X) are in the
ideal I(cid:48). Proposition 4.1.0.8 will easily follow. The first relation is more straightforward:

Lemma 4.1.1. The entries of X2 are in the ideal I(cid:48).

d(cid:88)

Proof. Let (zi,j)1≤i,j≤d := X2, where zi,j =

xi,axa,j. Now, set

a=1

ti,j := xi,jT r(X) ∈ I(cid:48).

Notice also that

a := xi,axa,j − xi,jxa,a ∈ I(cid:48)
si,j

26

from the minors relations. Therefore

ti,j +

d(cid:88)

a=1

a = zi,j ∈ I(cid:48).
si,j

We have to work harder in order to show that the entries of XtS0X − 2π(S0X + πS1X)
are in the ideal I(cid:48). The first step is as follows. By a simple direct calculation the relation

XtS1X + 2S0X + 2πS1X = 0 implies that:

E1 = −1
2
E2 = −1
2
E3 = −1
2
E4 = −1
2
O1 = −1
2
O2 = −1
2

Jn−rBt

2J2rB1,

Jn−rBt

2J2rB2,

Jn−rBt

1J2rB1,

Jn−rBt

1J2rB2,

Jn−rBt

2J2rA,

Jn−rBt

1J2rA.

(4.1.0.1)

(4.1.0.2)

(4.1.0.3)

(4.1.0.4)

(4.1.0.5)

(4.1.0.6)

Therefore, all the entries from Ei for i ∈ {1, 2, 3, 4} and O1, O2 can be expressed in terms of

the entries of B1, B2. The second step is the following lemma.

Lemma 4.1.2. Assume that all the 2 × 2 minors of the matrix X are 0. Then, the matrix

B1Jn−rBt

2 is symmetric.

Proof. Set (θij)1≤i,j≤2r := B1Jn−rBt

2. By direct calculations we find

θij =

xn−r+i,n−r−t+1xn−r+j,n+r+t.

n−r(cid:88)

t=1

27

So,

n−r(cid:88)

t=1

θji =

xn−r+j,n−r−t+1xn−r+i,n+r+t.

From the minor relations we have that xi,jxt,s = xi,sxt,j. Using this and the description of

the θij, θji we can easily see that θij = θji.

A useful observation, which will be used in the following lemma, is that the condition
∧2X = 0 together with the fact that the blocks B1, A, and B2, all share the same rows of

X, easily give

AB1 = Tr(A)B1, AB2 = Tr(A)B2.

(4.1.0.7)

We are now ready to show:
Lemma 4.1.3. The entries of XtS0X − 2π(S0X + πS1X) are in the ideal I(cid:48).

Proof. Using the block form of the matrix X and the relation XtS1X + 2(S0X + πS1X) = 0
modulo I(cid:48), it suffices to prove that:

(i) Et

1Jn−rE3 + Et

3Jn−rE1 − 2πJn−rE3 = 0,

(ii) Et

2Jn−rE4 + Et

4Jn−rE2 − 2πJn−rE2 = 0,

(iii) Et

1Jn−rE4 + Et

3Jn−rE2 − 2πJn−rE4 = 0,

(iv) Et

2Jn−rE3 + Et

4Jn−rE1 − 2πJn−rE1 = 0,

(v) Ot

1Jn−rE3 + Ot

2Jn−rE1 − 2π2J2rB1 = 0,

(vi) Ot

1Jn−rE4 + Ot

2Jn−rE2 − 2π2J2rB2 = 0,

(vii) Ot

1Jn−rO2 + Ot

2Jn−rO1 − 2π2J2rA = 0

28

in the quotient ring S/I(cid:48). We prove the first relation (i) and with the same arguments we can

prove the relations (ii)-(iv). Below we use the relations (1) and (3) for E1, E3 from above,

the relations B2Jn−rBt

1 = AJ2r, AB1 = T r(A)B1 and Lemma 4.1.2.

1Jn−rE3 + Et
Et

3Jn−rE1 − 2πJn−rE3

=

=

=

1
4
1
2
1
2

1J2rB2Jn−rBt
Bt

1J2rB1 +

1
4

1J2rB1Jn−rBt
Bt

2J2rB1 + πBt

1J2rB1

1J2rB2Jn−rBt
Bt

1J2rB1

1J2rB1 + πBt
1
2

1J2rB1 =

Bt
1J2rAB1 + πBt

T r(A)Bt

1J2rB1 + πBt

1J2rB1 = 0.

The last equality holds because T r(A) + 2π = 0.

Next, we prove the relation (v). The relations (vi), (vii) can be proved using similar

arguments. We use the relations (1), (3), (5) and (6) from above to express E1, E3, O1,

O2 in terms of B1 and B2. We use Lemma 4.1.2 and the relations B2Jn−rBt

1 = AJ2r and

AB1 = T r(A)B1.

1Jn−rE3 + Ot
Ot

2Jn−rE1 − 2π2J2rB1

1
4
1
2

=

AtJ2rB2Jn−rBt

1
4

AtJ2rB1Jn−rBt

1J2rB1 +
1J2rB1 − 2π2J2rB1 =

2J2rB1 − 2π2J2rB1
AtJ2rAB1 − 2π2J2rB1

1
2

AtJ2rB2Jn−rBt

=
= −πJ2rAB1 − 2π2J2rB1 = −π(T r(A)J2rB1 + 2πJ2rB1) = 0.

Proposition 4.1.0.8. We have I(cid:48) = I.

Proof. From Lemma 4.1.1 and Lemma 4.1.3 we get the desired result.

29

4.2 S/I(cid:48) (cid:39) S(cid:48)(cid:48)/I(cid:48)(cid:48).

The goal of this section is to prove that S/I(cid:48) is isomorphic to S(cid:48)(cid:48)/I(cid:48)(cid:48). Recall

I(cid:48) =

(cid:16) ∧2 X, T r(X), T r(A) + 2π, B2Jn−rBt
(cid:17)

XtS1X + 2(S0X + πS1X)

.

1 − AJ2r,

We first simplify and reduce the number of generators of I(cid:48). The desired isomorphism will

then follow.

Lemma 4.2.1. The trace T r(X) belongs to the ideal

(cid:16)∧2X, T r(A) + 2π, B2Jn−rBt

1 − AJ2r, XtS1X + 2(S0X + πS1X)

(cid:17)

.

Proof. We first write:

T r(X) = T r(E1) + T r(E4) + T r(A).

By the relations (1), (4) from Section 4.1 we get that the entries of E1 + 1

2 Jn−rBt

2J2rB1 and

E4 + 1

2 Jn−rBt

1J2rB2, belong to the ideal

(cid:16)∧2X, T r(A) + 2π, B2Jn−rBt

1 − AJ2r, XtS1X + 2(S0X + πS1X)

(cid:17)

.

Also, the element

T r(Jn−rBt

1J2rB2) − T r(A)

30

belongs to the above ideal. Thus,

T r(X) = T r(E1) + T r(E4) + T r(A)

= T r(E1 +

1
2

Jn−rBt

2J2rB1) + T r(E4 +

1
2

Jn−rBt

1J2rB2) + T r(A)

−1
2

T r(Jn−rBt

1J2rB2) − 1
2

T r(Jn−rBt

2J2rB1)
1
2

Jn−rBt

= T r(E1 +

1
2

Jn−rBt

2J2rB1) + T r(E4 +

1J2rB2),

belongs to the above ideal, as desired.

From the above lemma we obtain

(cid:16)∧2X, T r(A) + 2π, B2Jn−rBt

I(cid:48) =

1 − AJ2r, XtS1X + 2(S0X + πS1X)

(cid:17)

.

Next, we show:

Lemma 4.2.2. We have I(cid:48) = (cid:0)∧2X, T r(A) + 2π, B2Jn−rBt

1 − AJ2r

(cid:1) + I(cid:48), where I(cid:48) is

the ideal generated by the relations (1)-(6) from Section 4.1.

Proof. Using the block form of the matrix X and the relation XtS1X + 2(S0X + πS1X) = 0,

it suffices to prove that:

(a) AtJ2rB1 + 2πJ2rB1 = 0,

(b) AtJ2rB2 + 2πJ2rB2 = 0,

(c) AtJ2rA + 2πJ2rA = 0,

in the quotient ring of S by(cid:0)∧2X, T r(A) + 2π, B2Jn−rBt

1 − AJ2r

(cid:1) + I(cid:48).

31

We first discuss (a). Recall that A = B2Jn−rBt

1J2r, AB1 = T r(A)B1 and T r(A)+2π = 0.

Thus,

AtJ2rB1 + 2πJ2rB1 = J2rB1Jn−rBt

2J2rB1 + 2πJ2rB1

= J2rAB1 + 2πJ2rB1

= J2rT r(A)B1 + 2πJ2rB1 = 0.

Using similar arguments we can prove that the relations (b) and (c) hold.

The final step is to look more carefully at the minors that come from ∧2X.

Lemma 4.2.3. ∧2X ∈ I(cid:48) +(cid:0)∧2(B1(cid:124)B2), T r(A) + 2π, B2Jn−rBt

1 − AJ2r

(cid:1) .

Proof. In the proof, we use phrases like: “minors only from Ec”, “minors only from A and

B(cid:96)”, or “minors from A and Ec”. Let us explain what we mean by these terms. Consider

the minor

mi,j

t,s =

xi,j xi,s

 = xi,jxt,s − xi,sxt,j.

xt,j xt,s

When we say that “the minor comes only from Ec” we mean that all of the entries {xi,j, xt,s,
xi,s, xt,j} are entries of Ec for c ∈ {1, 2, 3, 4}. Similarly, when we say “the minor comes
only from A and B(cid:96)” we mean that all of {xi,j, xt,s, xi,s, xt,j} are entries either of A or
of B(cid:96) and at least one of the {xi,j, xt,s, xi,s, xt,j} is an entry of A and at least one of the
{xi,j, xt,s, xi,s, xt,j} is an entry of B(cid:96). On the other hand, when we say that “the minor
comes from A and Ec” we mean that at least one of the {xi,j, xt,s, xi,s, xt,j} is an entry of
A and at least one of the {xi,j, xt,s, xi,s, xt,j} is an entry of Ec for c ∈ {1, 2, 3, 4}. We have

the following cases of minors:

32

1. only from Ec

2. only from A

5. only from A and B(cid:96)

6. only from Ec and Om

3. only from Om

7. only from A and Om

4. from Ec and A

8. only from Ec and B(cid:96)

In each case, we will show that the corresponding minors belong to

(cid:16)∧2(B1(cid:124)B2), T r(A) + 2π, B2Jn−rBt

I(cid:48) +

(cid:17)

.

1 − AJ2r

We will start by considering case (1), i.e. minors only from Ec. It suffices to prove xi,jxt,s =

xi,sxt,j in the quotient ring

I(cid:48) +(cid:0)∧2(B1(cid:124)B2), T r(A) + 2π, B2Jn−rBt

S

1 − AJ2r

(cid:1) .

By using minors from B(cid:96) for (cid:96) ∈ {1, 2} and for all i, j ∈ Zc, we have the following equation

in the above quotient ring:

(cid:16)
(cid:16)

n(cid:88)
n(cid:88)

a=n−(r−1)

a=n−(r−1)

(cid:17)(cid:16)
(cid:17)(cid:16)

n(cid:88)
n(cid:88)

a=n−(r−1)

a=n−(r−1)

(cid:17)
(cid:17)

xd+1−a,d+1−ixa,j

xd+1−a,d+1−txa,s

=

xd+1−a,d+1−ixa,s

xd+1−a,d+1−txa,j

.

By using the relations (1)-(4) from Section 4.1 we can express the entries xi,j of Ec as:

xi,j = −

a=n−(r−1)

xd+1−a,d+1−ixa,j

n(cid:88)

33

with i, j ∈ Zc. Using this and the above equality we obtain:

xi,jxt,s = xi,sxt,j.

The rest of cases (2)-(8) can be handled by similar arguments. More precisely, by using the
relations (1)-(6) from Section 4.1 we can express all the entries from Ei for i ∈ {1, 2, 3, 4} and
O1, O2 in terms of the entries of B1, B2. Also, by using A = B2Jn−rBt
1J2r we can express
all the entries of A in terms of the entries of B1, B2. After that, by using the 2 × 2-minors
from the matrix (B1|B2) we get the desired result in all the remaining cases.

End of proof of Theorem 4.0.1: From the above lemma we obtain that

Observe that an equivalent way of writing I(cid:48) is:

(cid:17)

1 − AJ2r

.

I(cid:48) = I(cid:48) +

(cid:16)∧2(B1(cid:124)B2), T r(A) + 2π, B2Jn−rBt
(cid:16)∧2(B1(cid:124)B2), T r(B2Jn−rBt

I(cid:48) = I(cid:48) +

1J2r) + 2π, B2Jn−rBt

1 − AJ2r

(cid:17)

.

Using this and the fact that I(cid:48) = I the proof of Theorem 4.0.1 follows.

34

Chapter 5

Flatness of Ud,l

We continue to assume d = 2n, l = 2r.

Recall that Ud,l = Spec (S/I). The goal of this chapter is to prove that Ud,l is flat over O

as given by Theorem 5.0.4. The simplification that we obtained from Theorem 4.0.1 quickly

gives the following, which in turn plays a crucial role for the proof of Theorem 5.0.4.

Theorem 5.0.1. Ud,l is Cohen-Macaulay.

Proof. Denote by O[B1|B2] the polynomial ring over O with variables the entries of the
matrix (B1|B2). From Theorem 4.0.1 we obtain the isomorphism

(cid:39)

S
I

(cid:0)∧2(B1(cid:124)B2), T r(B2Jn−rBt

O[B1|B2]

1J2r) + 2π(cid:1) .

Set R := O[B1|B2]/(∧2 (B1(cid:124)B2)). By [21, Remark 2.12], the ring R is Cohen-Macaulay and

an integral domain. We consider the point P of the determinantal variety which is defined

by the relations:

xn−r+1,1 xn−r+1,d

 =

1 − π 1 − π



xn+r,1

xn+r,d

1

1

and we set all the other variables equal to zero. We can easily observe that T r(B2Jn−rBt

1J2r)+

35

2π is not zero over the point P . Therefore, we have

ht((T r(B2Jn−rBt

1J2r) + 2π)) = 1.

We apply the fact that if A is Cohen-Macaulay and the ideal I = (a1, . . . , ar) of A
has height r then A/I is Cohen-Macaulay ([12, example 3 (16.F)]) to A = R and a1 =
T r(B2Jn−rBt

1J2r) + 2π. We obtain that

(cid:0)∧2(B1(cid:124)B2), T r(B2Jn−rBt

O[B1|B2]

1J2r) + 2π(cid:1)

is Cohen-Macaulay. This implies the result.

Remark 5.0.2. From the above proof and the standard formula for the dimension of the
determinantal varieties (see [21, Proposition 1.1]) we obtain that dim(S/I) = dim(R) − 1 =
d − 1. Hence, the dimension of Ud,l is d − 1.

Remark 5.0.3. Mimicking the proof of Theorem 5.0.1 and by considering Remark 5.0.2 we
obtain that the special fiber U d,l of Ud,l is Cohen Macaulay and of dimension d − 2.

Theorem 5.0.4. Ud,l is flat over O.

Proof. From Remark 5.0.2 we have that Ud,l has dimension d − 1. From Remark 5.0.3 we
have that the dimension of U d,l is d − 2. Using the fact that Ud,l is Cohen Macaulay, see

Theorem 5.0.1, we obtain that ht((π)) = 1. Hence, we have that (π) is a regular sequence

i.e π is not a zero divisor (see [9]). From this, flatness of Ud,l follows.

36

Chapter 6

Reducedness of U d,l

In all of Chapter 6, we assume d = 2n and l = 2r.

We will prove that the special fiber U d,l of Ud,l = Spec (S/I) is reduced; see Chapter 4

for undefined terms.

Proof of Theorem 3.2.3. From Remark 5.0.3 we know that U d,l is Cohen-Macaulay. By

using Serre’s criterion for reducedness, it suffices to prove that the localizations at minimal

primes are reduced. From Lemma 7.1.1 and Lemma 7.2.1 we obtain the minimal primes of
U d,l. Below, we focus on the localization of U d,l over I1 for 1 < r < n − 1 (see Section 7.2

for the notation). In the other cases the proof is similar.

We first introduce some additional notation:
Denote by k[B1|B2] the polynomial ring over k with variables the entries of the matrix
(B1|B2). We set Z1 := Z \ {n − r + 1} and Z2 := Zc \ {1}. By direct calculations we get

T r(B2Jn−rBt

1Jl) =

xi,j xd+1−i,d+1−j.

(cid:88)

n−r+1≤i≤n+r,1≤j≤n−r

Set tr := 1

2 T r(B2Jn−rBt

1Jl) and t(cid:48)

r = tr − xn−r+1,1xn+r,d. Lastly, we set m := x−1

n−r+1,dt(cid:48)
r.

We refer the reader to Section 7.2 for the rest undefined terms.

From Theorem 4.0.1 and Lemma 4.1.2 we obtain that the special fiber U d,l is given by

37

the quotient of k[B1|B2] by the ideal (∧2(B1(cid:124)B2), tr). Set J1 = I1 +(cid:0)∧2(B1(cid:124)B2), tr

(cid:1) where

(cid:16)

n(cid:88)

I1 =

(

a=n−(r−1)

xd+1−a,d+1−ixa,j)i,j∈Zc, ∧2(B1(cid:124)B2)

(cid:17)

.

By localizing k[B1|B2]/(∧2(B1(cid:124)B2), tr) at J1 we have
(cid:0)∧2(B1(cid:124)B2), tr

(cid:18) k[B1|B2]
∧2(B1(cid:124)B2), tr

k[B1|B2]I1

(cid:19)

(cid:39)

J1

(cid:1)

I1

.

In the proof of Lemma 7.2.1 we used the fact that xn−r+1,1 /∈ I1. Similarly, we have that
xn−r+1,d /∈ I1.

Claim:

(cid:16)

=

(cid:16)∧2(B1(cid:124)B2), tr

(cid:17)

I1

(xi,j − x−1

n−r+1,1xi,1xn−r+1,j)i∈Z1, j∈Z2

(cid:17)

.

I1

, (xn+r,1 + m)

Proof of the claim: From the minors we have xi,jxn−r+1,1−xi,1xn−r+1,j = xn−r+1,1(xi,j−
x−1
n−r+1,1xi,1xn−r+1,j). We rewrite tr as:

tr = xn−r+1,1xn+r,d + t(cid:48)
r.

(6.0.0.1)

Combining (6.0.0.1) with the minor xn−r+1,1xn+r,d = xn−r+1,dxn+r,1 we obtain

xn−r+1,1xn+r,d + t(cid:48)

r = xn−r+1,d(xn+r,1 + m).

Now, because xn−r+1,1, xn−r+1,d /∈ I1 the claim follows.

38

Combining all the above we have:

(cid:1)

I1

k[B1|B2]I1

(cid:0)∧2(B1(cid:124)B2), tr
(cid:16)
n−r+1,1xi,1xn−r+1,j)i∈Z1, j∈Z2
(cid:39) k[(xi,1)i∈Z\{n+r}, (xn−r+1,j)j∈Zc, x−1

k[B1|B2]I1

(xi,j − x−1

(cid:39)

(cid:17)

, (xn+r,1 + m)

n−r+1,1, x−1

I1
n−r+1,d]J1

where the last one is a reduced ring. Thus, the special fiber of Ud,l is reduced.

39

Chapter 7

Irreducible Components of U d,l Part I

We continue to assume d = 2n, l = 2r.

Recall that Ud,l = Spec (S/I) and U d,l is the special fiber of Ud,l. In this chapter, the

main goal is to calculate the irreducible components of U d,l.

Theorem 7.0.1. (i) For r = 1 and r = n − 1, U d,l has three irreducible components.

(ii) For 1 < r < n − 1, U d,l has two irreducible components.

The proof of the theorem will be carried out in Section 7.1 (case (i)) and in Section 7.2

(case (ii)).

7.1

l = 2 and l = d − 2

In this section we will prove Theorem 7.0.1 in the case r = 1. A similar argument works in
the case r = n − 1. For this section we introduce the following notation. Observe that when
r = 1, Z = {n, n + 1} and Zc = {1, 2, 3, . . . , d} \ Z. We rename the variables as follows:

vi = xn,i, for i ∈ Zc and wj = xn+1,j, for j ∈ Zc.

Define the polynomial ring Ssim = k[vi, wj]i,j∈Zc. From Theorem 4.0.1 we obtain that the

40

special fiber is isomorphic to Ssim/Isim where

(cid:16) n−1(cid:88)

i=1

Isim =

viw2n+1−i,

(viwj − vjwi)i,j∈Zc

It is not very hard to observe, by using the minors, that

(cid:17)

.

n−1(cid:88)

j=1

i=1

Next, we set

Isim =

viw2n+1−i, (vi

wjw2n+1−j)i∈Zc, (wi

vjv2n+1−j)i∈Zc,

(cid:16) n−1(cid:88)
n−1(cid:88)

i=1

n−1(cid:88)

j=1

wjw2n+1−j,

(cid:17)

,

n−1(cid:88)
(cid:16)

j=1

viv2n+1−i

I1 =

(cid:17)

.

(viwj − vjwi)i,j∈Zc
(cid:16)

(cid:17)

(vi)i∈Zc

I2 =

(wi)i∈Zc

and

(cid:16) n−1(cid:88)

i=1

I3 =

viv2n+1−i,

n−1(cid:88)

i=1

wiw2n+1−i,

n−1(cid:88)

i=1

viw2n+1−i, (viwj − vjwi)i,j∈Zc

(cid:17)

.

Proof of Theorem 7.0.1 (i): From the above, it suffices to calculate the irreducible components

of V (Isim).

Observe that the elements

(vi

belong to Isim.

n−1(cid:88)

j=1

vjv2n+1−j)i∈Zc,

wjw2n+1−j

n−1(cid:88)

wjw2n+1−j)i∈Zc,

(wi

j=1

n−1(cid:88)

i=1

viv2n+1−i

n−1(cid:88)

j=1

41

Therefore, we can easily see that

V (Isim) = V (I1) ∪ V (I2) ∪ V (I3).

Observe also that

Ssim/I1 (cid:39) k[(wj)j∈Zc],

Ssim/I2 (cid:39) k[(vj)j∈Zc].

Thus, the closed subschemes V (I1), V (I2) are affine spaces of dimension d−2 and so they are

irreducible and smooth. We have to check that the third one is irreducible and of dimension
d − 2. Notice that I3 is generated by homogeneous elements. Thus, it suffices to prove that

the projectivization

Vp(I3) ⊆ P2d−5

k

of the affine cone V (I3) is irreducible. Consider

Vv1 := Vp(I3) ∩ Uv1,

where Uv1 = D(v1 (cid:54)= 0). We can see that it is isomorphic to

Spec (k[(vi)i∈Zc\{1,d}, w1]),

and so it is irreducible. By symmetry we have a similar result for every Vvi and Vwj with
i, j ∈ Zc. Moreover, the Vvi and Vwj form a finite open cover of irreducible open subsets of
Vp(I3). Thus Vp(I3) is irreducible and so V (I3) ⊆ A2d−4

is irreducible. This completes the

k

proof of Theorem 7.0.1 (i).

42

We can go one step further and prove that:

Lemma 7.1.1. The ideals I1, I2, I3 are prime.

Proof. From the proof of Theorem 7.0.1 (i), I1, I2 are clearly prime ideals. It suffices to prove

that I3 is prime. From Theorem 7.0.1 (i) we have that I3 is a primary ideal and so every zero

divisor in D := k[vi, wj]i,j∈Zc/I3 is a nilpotent element. Hence, it suffices to prove that D is

reduced. From the proof of Theorem 7.0.1 (i) we have that the scheme Spec (D) is smooth

over Spec (k) outside from its closed subscheme of dimension 0 which is defined by the ideal

((vj)j∈Zc, (wi)i∈Zc). Therefore, using Serre’s criterion for reducedness ([12] 17.I) and the
above description it suffices to find a regular element f such that f ∈ ((vj)j∈Zc, (wi)i∈Zc).

Claim: We can take f = w1.

Proof of the claim: Assume that w1 is not a regular element. Then, because I3 is primary,
1 ∈ I3 for some m > 0. We will obtain a contradiction by using Buchberger’s

we have wm

algorithm. This is a method of transforming a given set of generators for a polynomial ideal

into a Gr¨obner basis with respect to some monomial order. For more information about this

algorithm we refer the reader to [4, Chapter 2].

Set R = k[vi, wj]i,j∈Zc for the polynomial ring. We choose the following order for our

variables:

v1 > v2 > ··· > vd > w1 > ··· > wd.

Then, the graded lexicographic order induces an order of all monomials in R.

Next, we recall the division algorithm in R. We fix the above monomial ordering. Let
J = (f1, . . . , fs) be an ordered s-tuple of polynomials in R. Then every g ∈ R can be written

as

g = a1f1 + . . . + asfs + r,

43

where ai, r ∈ R and either r = 0 or r is a linear combination, with coefficients in k, of

monomials, none of which is divisible by any of LT (f1), . . . , LT (fs). By LT (fi) we denote

the leading term of fi. We will call r a remainder of g on division by J. (See [4, Chapter 2]

for more details.)

Recall that the S-polynomial of the pair f , g ∈ R is

S(f, g) =

LCM (LM (f ), LM (g))

LT (f )

f − LCM (LM (f ), LM (g))

LT (g)

g.

Here, by LM (f ) we denote the leading monomial of f according to the above ordering.

To find the Gr¨obner basis for the ideal I3, we start with the generating set

(cid:110) n−1(cid:88)

n−1(cid:88)

n−1(cid:88)

(cid:111)

.

viv2n+1−i,

wiw2n+1−i,

viw2n+1−i, (viwj − vjwi)i,j∈Zc

i=1

i=1

i=1

Then, we calculate all the S-polynomials S(f, g), where f, g are any two generators from the
generating set that we have started. If all the S-polynomials are divisible by the generating

set then the generating set already forms a Gr¨obner basis. On the other hand, if a remainder

is nonzero we extend our generating set by adding this remainder and we repeat the above
process until we have a generating set where all the S-polynomials are divisible by the

generating set. In our case, the generators of I3 are homogeneous polynomials of degree 2.

The monomials of those homogeneous polynomials have one of the following forms:

1. vivj with i (cid:54)= j, or

2. wiwj with i (cid:54)= j, or

3. wivj with i (cid:54)= j.

44

Thus, the nonzero remainder of any S-polynomial is a polynomial where each monomial is

divisible by at least one monomial of the above three forms. By this observation we can

see that, the Gr¨obner basis cannot have a monomial that looks like wm
{g1 . . . , gN} be the Gr¨obner basis of I3. From the above we have

i or vm

j . Now, let

1 /∈ (cid:104)LT (g1), . . . , LT (gN )(cid:105).
wm

Moreover, because {g1 . . . , gN} is the Gr¨obner basis of I3 we have that

(cid:104)LT (g1), . . . , LT (gN )(cid:105) = LT (I3),

(see [4, Chapter 5]). By LT (I3) we denote the ideal generated by all the leading terms of

elements of I3. Therefore,

1 /∈ LT (I3).
wm

Hence, w1 is a regular element and so I3 is a prime ideal. This completes the proof of the

claim and by the above the proof of lemma.

7.2

2 < l < d − 2

In this section we will prove Theorem 7.0.1 in the case 1 < r < n − 1. In this case we have
Z = {n− (r − 1), . . . , n, n + 1, . . . , d− n + r} and Zc = {1, 2, 3, . . . , d}\ Z. For the undefined

terms below we refer the reader to Chapters 4 and 6. From Theorem 4.0.1 we obtain that

the special fiber U d,l is given by the quotient of k[B1|B2] by the ideal Is = (∧2(B1(cid:124)B2), tr).

45

Also with direct calculations and by using the minors we can see that

Is =

tr,

(cid:16) ∧2 (B1(cid:124)B2),
n−r(cid:88)

(cid:17)

.

(cid:17)

i,j∈Z
t,s∈Zc

(cid:16)

a=n−(r−1)

n(cid:88)
(cid:16)(cid:16)

I2 =

n(cid:88)
(cid:16)(cid:16) n−r(cid:88)

a=1

I1 =

a=n−(r−1)

xd+1−a,d+1−txa,s

xi,bxd+1−j,d+1−b

b=1

(cid:17)

xd+1−a,d+1−txa,s

(cid:17)

i,j∈Z

xi,axd+1−j,d+1−a

t,s∈Zc

(cid:17)
, ∧2(B1(cid:124)B2)
(cid:17)
, ∧2(B1(cid:124)B2)

.

Next, set

and

Proof of Theorem 7.0.1 (ii): From the above, it suffices to calculate the irreducible compo-

nents of V (Is).

Observe that

(cid:16)

n(cid:88)

a=n−(r−1)

xd+1−a,d+1−txa,s

n−r(cid:88)

b=1

xi,bxd+1−j,d+1−b

(cid:17)

∈ Is.

i,j∈Z
t,s∈Zc

Therefore, we can easily see that

V (Is) = V (I1) ∪ V (I2).

Next, we prove that the closed subschemes V (I1) and V (I2) are irreducible of dimension
d − 2. We will start by proving that V (I1) is an irreducible component. Observe that I1 is

generated by homogeneous elements.

46

Thus, it suffices to prove that the projectivization

Vp(I1) ⊆ P4r(n−r)−1

k

.

of the affine cone V (I1) is irreducible. We look at

Vxn−(r−1),1

:= Vp(I1) ∩ Uxn−(r−1),1

,

where Uxn−(r−1),1

= D(xn−(r−1),1 (cid:54)= 0). We can see that it is isomorphic to

Spec (k[(xn−(r−1),j)j∈Zc\{1}, (xt,1)n−(r−2)≤t≤n+r−1])

and so it is irreducible. Because of the symmetry of the relations we will have a similar result

for every Vxt,s with t ∈ Z, s ∈ Zc. Moreover,

(Vxt,s)t∈Z, s∈Zc

form a finite open cover of irreducible open subsets of Vp(I1) and thus we get that Vp(I1) is
irreducible and so V (I1) is irreducible of dimension d − 2.

We use a similar argument for V (I2). I2 is generated by homogeneous elements and so

by using the projectivization

Vp(I2) ⊆ P4r(n−r)−1

k

of the affine cone V (I2), it suffices to prove that Vp(I2) is irreducible. We look at

Vxn−(r−1),1

:= Vp(I2) ∩ Uxn−(r−1),1

,

47

where Uxn−(r−1),1

= D(xn−(r−1),1 (cid:54)= 0). We can see that it is isomorphic to

Spec (k[(xn−(r−1),j)j∈Zc\{1,d}, (xt,1)n−(r−2)≤t≤n+r])

and so it is irreducible. Therefore, V (I2) is irreducible of dimension d − 2. This completes

the proof of Theorem 7.0.1 (ii).

Next, we prove that:

Lemma 7.2.1. The ideals I1, I2 are prime.

Proof. To see that I1, I2 are prime ideals one proceeds exactly as in Lemma 7.1.1. So, it
suffices to find a regular element f such that f ∈ ((xt,s)t∈Z,s∈Zc). We claim that xn−r+1,1
is a choice for f . Assume for contradiction that xn−r+1,1 is not a regular element. Then,

because Ii is primary, we should have that xm

n−r+1,1 ∈ Ii for some m > 0.

We choose the following order for our variables (xt,s)t∈Z,s∈Zc:

xn−r+1,1 > . . . > xn−r+1,n−r > xn−r+1,n+r+1 > . . . > xn−r+1,d >

xn−r+2,1 > . . . > xn−r+2,d > . . . > xn+r,1 > . . . > xn+r,n−r >

xn+r,n+r+1 > . . . > xn+r,d.

Then, the graded lexicographic order induces an ordering to all the monomials. First, let’s

consider the ideal I1. In order to find the Gr¨obner basis for I1, we start with the generating

set

(cid:110)(cid:16)

n(cid:88)

a=n−(r−1)

xd+1−a,d+1−ixa,j

(cid:17)

(cid:111)

.

i,j∈Zc

, ∧2(B1(cid:124)B2)

After that we calculate all the S-polynomials S(f, g), where f, g are any two generators

48

from the generating set that we have started; so in our case is I1. The generators of I1 are

homogeneous polynomials of degree 2. The monomials of those homogeneous polynomials

have one of the following form:

(xi,jxt,s)

with i, t ∈ Z, j, s ∈ Zc and either i (cid:54)= t or j (cid:54)= s. Thus, the nonzero remainder of any
S-polynomial is a polynomial where each monomial is divisible by at least one monomial

of the above form. By this observation we can see that, the Gr¨obner basis cannot have a

monomial that looks like xm

n−r+1,1. Now, by using a Gr¨obner basis argument as in Lemma

7.1.1 we deduce that xn−r+1,1 is a regular element and so I1 is a prime ideal.

Now, by looking the ideal I2 we have the generating set:

(cid:110)(cid:16) n−2(cid:88)

a=1

xi,axd+1−j,d+1−a

i,j∈Z

(cid:17)

(cid:111)
, ∧2(B1(cid:124)B2)

.

So, we observe that in this case also the generators of the ideal I2 are homogeneous polyno-

mials of degree 2. All the monomials have one of the following form:

(xi,jxt,s)

with i, t ∈ Z, j, s ∈ Zc and either i (cid:54)= t or j (cid:54)= s. So, by using the same argument we can

prove that I2 is a prime ideal.

49

Chapter 8

The Remaining Cases

In this chapter we sketch the proof of Theorems 3.2.1 and 3.2.2 in the remaining cases of

parity for d and l. The main point is that in all the cases the affine chart Ud,l of the local

model is displayed as a hypersurface in a determinantal scheme. The arguments are similar

with the proof of the case (d, l) = (even, even). In fact, in the case that (d, l) = (odd, odd)

the argument is exactly the same. The case of Theorem 3.2.2 (different parity) is somewhat

different as we explain below.

8.1 Proof of Theorem 3.2.2

Proof. We use the notation from Section 3.2. We also introduce some new notation. Set

Z(cid:48) := {n − r(cid:48), . . . , n, n + 2, . . . , d − n + r(cid:48)},

(Z(cid:48))c := {1, 2, 3, . . . , d} \ Z(cid:48).

Also, define the polynomial ring

O[B(cid:48)

1|Q(cid:48)|B(cid:48)

2] := O[(xt,s)t∈Z(cid:48),s∈(Z(cid:48))c].

Lastly, set

∧2(B(cid:48)
1(cid:124)Q(cid:48)

(cid:124)B(cid:48)
2) := (xi,jxt,s − xi,sxt,j)i,t∈Z(cid:48) j,s∈(Z(cid:48))c.

50

We can now sketch the proof. In this case, similar elementary arguments as in the proof of
Theorem 4.0.1 give that the quotient O[X]/I is isomorphic to the quotient of O[B(cid:48)

1|Q(cid:48)|B(cid:48)
2]

by the ideal

1(cid:124)Q(cid:48)
(∧2(B(cid:48)

2), T r((B(cid:48)
(cid:124)B(cid:48)

2Jn−r(cid:48)(B(cid:48)

1)t +

Q(cid:48)(Q(cid:48))t)Jl) + 2π).

1
2

The rest of the argument deducing flatness is the same as before.

51

Chapter 9

Irreducible Components of U d,l Part II

In this chapter we present the irreducible components of the special fiber U d,l of Ud,l in the

remaining cases. We omit the proofs which are similar to Theorem 7.0.1. For the notation

we refer the reader to Chapter 4 and Chapter 8.

9.1 (d, l) = (odd, odd)

9.1.1

When l < d − 2, the irreducible components of U d,l are the closed subschemes V (I1) and

V (I2) where:

I1 =

(cid:16)

n(cid:88)

(
a=n−r+1

xd+1−a,d+1−txa,s +

1
2

xn+1,d+1−txn+1,s)t,s∈Zc, ∧2(B1(cid:124)B2)

(cid:17)

(cid:16)

(

n−r(cid:88)

a=1

I2 =

xi,axd+1−j,d+1−a)i,j∈Z , ∧2(B1(cid:124)B2)

(cid:17)

.

and

9.1.2

When l = d − 2, the irreducible components of U d,l are the closed subschemes V (I1), V (I2)

52

and V (I3) where:

and

(cid:16)

n(cid:88)

a=n−r+1

I3 =

(cid:16)

(xi,1)i∈Z

(cid:17)

,

(cid:16)

I2 =

(xi,d)i∈Z

(cid:17)

I1 =

n(cid:88)

a=n−r+1

xa,1xd+1−a,1 +

1
2

x2
n+1,1,

xa,dxd+1−a,d +

1
2

x2
n+1,d,

n(cid:88)

a=n−r+1

∧2(B1(cid:124)B2),

xa,1xd+1−a,d +

1
2

xn+1,1xn+1,d

(cid:17)

.

9.2 (d, l) = (odd, even)

9.2.1

When l > 2 the irreducible components of U d,l are the closed subschemes V (I1) and V (I2)

where:

I1 =

(cid:17)

(cid:16)(cid:16)

n(cid:88)
(cid:16)

a=n−(r−1)

n(cid:88)

xd+1−a,d+1−txa,s

t,s∈(Z(cid:48))c

2),

(cid:124)B(cid:48)

, ∧2((B(cid:48)
1(cid:124)Q(cid:48)
(cid:17)

(cid:17)

xd+1−a,n+1xa,j

1≤j≤d

a=n−(r−1)

(cid:16) ∧2 ((B(cid:48)
1(cid:124)Q(cid:48)

2),

(cid:124)B(cid:48)

(cid:16) n−r(cid:88)

a=1

I2 =

and

9.2.2

xi,axd+1−j,d+1−a +

1
2

xi,n+1xd+1−j,n+1

(cid:17)

i,j∈Z(cid:48)

(cid:17)

.

When l = 2, the irreducible components of U d,l are the closed subschemes V (I1), V (I2) and

V (I3) where:

I1 =

(cid:16)

(xn,i)i∈(Z(cid:48))c

(xn+2,i)i∈(Z(cid:48))c

(cid:17)

(cid:17)

(cid:16)

,

I2 =

53

and

(cid:16) n−1(cid:88)

a=1

I3 =

n−1(cid:88)

a=1

xn,axn,d+1−a +

1
2

x2
n,n+1,

xn+2,axn+2,d+1−a +

1
2

x2
n+2,n+1,

n−1(cid:88)

a=1

1(cid:124)Q(cid:48)
∧2((B(cid:48)

2),

(cid:124)B(cid:48)

xn,axn+2,d+1−a +

1
2

xn,n+1xn+2,n+1

(cid:17)

.

9.3

(d, l) = (even, odd)

The irreducible components of U d,l are the closed subschemes V (I1) and V (I2) where:

1
2

1
2

54

a=n−r

(cid:16)(cid:16) n−1(cid:88)
(cid:16) n−1(cid:88)
(cid:16)(cid:16) n−r−1(cid:88)
(cid:16) n−r−1(cid:88)

a=n−r

a=1

a=1

I1 =

xd+1−a,n+1xa,j +

1
2

xn,n+1xn,j

1≤j≤d

xd+1−a,d+1−txa,s +

1
2

xn,d+1−txn,s

I2 =

xi,axn,d+1−a +

xi,n+1xn,n+1

xi,axd+1−j,d+1−a +

xi,n+1xd+1−j,n+1

(cid:17)

(cid:17)

2),

(cid:124)B(cid:48)

, ∧2((B(cid:48)
1(cid:124)Q(cid:48)
(cid:17)

(cid:17)

t,s∈(Z(cid:48))c

,

2),

i∈Z(cid:48), ∧2((B(cid:48)
1(cid:124)Q(cid:48)
(cid:17)

(cid:124)B(cid:48)
(cid:17)

i∈Z(cid:48),j∈Z(cid:48)\{n}

.

Chapter 10

The Blow-Up of Mloc(Λ)

The statements and the results from this chapter are contained in [18]. The reader is referred

to loc. cit. §5.3 for more details.

Let

rbl : Mbl(Λ) → Mloc(Λ)

be the blow-up of Mloc(Λ) at the closed point ∗ of its special fiber that corresponds to L = L.

We will show:

Theorem 10.0.1. The scheme Mbl(Λ) is regular and has special fiber a divisor with normal
crossings. In fact, Mbl(Λ) is covered by open subschemes which are smooth over Spec (Zp[u, x,
y]/(u2xy − p)).

Before we start the proof of the above theorem, we first restate Theorems 3.2.1 and 3.2.2

in a different form; see Theorem 10.0.2. Then using this we prove Theorem 10.0.1.

In this chapter we use the notation from Section 3.2 and we also introduce some additional

notation: we set

T(B(cid:48)

1|Q(cid:48)|B(cid:48)

2) = Tr((B(cid:48)

T(B1|B2) = Tr(B2Jn−rBt
2Jn−r(cid:48)(B(cid:48)

1Jl),
Q(cid:48)(Q(cid:48))t)Jl),

1)t +

1
2

if d ≡ l mod 2,
if d (cid:54)≡ l mod 2.

55

For simplicity, we define

Z =

[B1|B2],

[B(cid:48)
1|Q(cid:48)|B(cid:48)

if d ≡ l mod 2,

2], if d (cid:54)≡ l mod 2.

Then Z = (zij) ∈ Matl×(d−l), in both cases. By an explicit calculation, we find that

(cid:88)

T(Z) =

1
2

1≤i≤l,1≤j≤d−l

zi d−l+1−j zl+1−i j.

l×(d−l) = {Z | ∧2Z =
(The same expression is valid for any pair (d, l).) Finally, denote by D 2
0} ⊂ Matl×(d−l) the “determinantal” subscheme of the affine space of matrices Z over
Spec (O).

Using the above notation, Theorems 3.2.1 and 3.2.2 are equivalent to the statement:

Theorem 10.0.2. An affine chart of the local model Mloc(Λ) around the worst point L = L

is given by Ud,l and is isomorphic to the closed subscheme DT of the determinantal scheme

D 2

l×(d−l) which is defined by the quadratic equation

(cid:88)

1≤i≤l,1≤j≤d−l

zi d−l+1−j zl+1−i j = −4π.

Proof. This follows from the proofs of Theorems 3.2.1 and 3.2.2. More precisely, see Theorem

4.0.1 and Chapter 8.

Now, we are ready to prove Theorem 10.0.1.

Proof. By Theorem 10.0.2, it is enough to show the conclusion of the theorem for the

blow-up (cid:101)DT of DT at the (maximal) ideal given by (zij). For simplicity, we write D for

56

the determinantal scheme D 2

l×(d−l) over Spec (O). This is the affine cone over the Segre

embedding

Also, we set

(Pl−1 × Pd−l−1)O (cid:44)→ Pl(d−l)−1

O

.

(cid:88)

1≤i≤l,1≤j≤d−l

T =

1
2

zi d−l+1−j zl+1−i j.

Let us consider the blow-up

(cid:101)D −→ D

of the determinantal scheme over Spec (O) along the vertex of the cone, i.e. along the

subscheme defined by the ideal (zij). Then, the blow-up (cid:101)DT is isomorphic to the strict
transform of the hypersurface DT ⊂ D given by T + 2π = 0. Let Vs,t be the open affine
chart of (cid:101)D over which the image of zst generates the pull-back of the ideal (zij). Then

Vs,t = Spec (O[(ui,j)1≤i≤l,1≤j≤d−l]/((ui,j − us,jui,t)i,j, us,t − 1).

The intersection Vs,t ∩ (cid:101)DT is obtained by substituting zij = ui,jzst and ui,j = us,jui,t, for

all i, j, in the equation T = −2π. This amounts to setting

zij = us,jui,tzst

and gives

(cid:88)

1≤i≤l,1≤j≤d−l

4π + z2

st(

us,d−l+1−jui,tus,jul+1−i,t) = 0.

57

(cid:88)

i(cid:54)=s

(cid:88)

j(cid:54)=t

us,jus,d−l+1−j.

This is

l(cid:88)

4π + z2

st(

d−l(cid:88)

ui,tul+1−i,t)(

us,jus,d−l+1−j) = 0.

(10.0.0.1)

i=1

j=1

Note that, since us,t = 1, the two sums in the line above are

S1 = ul+1−s,t +

ui,tul+1−i,t ,

S2 = us,d−l+1−t +

We see that u (cid:55)→ zst, x (cid:55)→ −S1/2, y (cid:55)→ S2/2 defines a smooth morphism

Vs,t ∩ (cid:101)DT −→ Spec (O[u, x, y]/(u2xy − π)).

58

Chapter 11

Application to Shimura Varieties

In this chapter we present one of the main results from the joint work [18]. For a more

detailed presentation we refer the reader to loc. cit. §7.

The goal is to construct regular integral models for GSpin Shimura varieties; see Theorem

11.0.1.

We start with an odd prime p and an orthogonal quadratic space V over Q of dimension
d ≥ 5 and signature (d − 2, 2). We take G = GSpin(V ) and we consider the hermitian

symmetric domain

D = {z ∈ VC : (cid:104)z, z(cid:105) = 0, (cid:104)z, ¯z(cid:105) < 0}/C×

of dimension d − 2. The pair (G, D) defines the spin similitude Shimura datum (for more
details see [18, §7.1].).

In addition, we choose a vertex lattice Λ ⊂ V ⊗Q Qp with πΛ∨ ⊂ Λ ⊂ Λ∨ and l =
lengthZp(Λ∨/Λ), l∗ = min(l, d − l), and assume l∗ ≥ 2. This defines the parahoric subgroup

Kp = {g ∈ GSpin(V ⊗Q Qp) | gΛg−1 = Λ, η(g) ∈ Z×
p }

(Here, η : GSpin(V ⊗Q Qp) → Q×

which we fix below.
p is the spinor similitude, and
for v ∈ V ⊗Q Qp, gvg−1 is defined using the Clifford algebra, see [18, §2.3, §2.5].) The
group G is the smooth connected “Bruhat-Tits” group scheme over Spec (Zp) such that

59

Qp = G ⊗Q Qp and G(Zp) = Kp. Choose also a sufficiently small compact open

G ⊗Zp
subgroup Kp of the prime-to-p finite adelic points G(Ap

f ) of G and set K = KpKp. The

Shimura variety ShK (G, D) with complex points

ShK (G, D)(C) = G(Q)\D × G(Af )/K

is of Hodge type and has a canonical model over the reflex field Q.

Theorem 11.0.1. For every Kp as above, there is a scheme S reg

K (G, D), flat over Spec (Zp),

with

K (G, D) ⊗Zp
S reg

Qp = ShK (G, D) ⊗Q Qp,

and which supports a “local model diagram”

π

reg
K

S reg

K (G, D)

such that:

(cid:102)S reg

K (G, D)

reg
q
K

Mbl(Λ)

(11.0.0.1)

a) πreg

K is a G-torsor for the parahoric group scheme G that corresponds to Kp,
K is smooth and G-equivariant.

b) qreg

c) S reg

K (G, D) is regular and has special fiber which is a divisor with normal crossings.

K (G, D) can be covered, in the ´etale topology, by schemes which are smooth

In fact, S reg
over Spec (Zp[u, x, y]/(u2xy − p)).

In addition, we have:

60

1) The schemes {S reg

K (G, D)}Kp, for variable Kp, support correspondences that extend
the standard prime-to-p Hecke correspondences on {ShK (G, D)}Kp. These correspon-
dences extend to the local model diagrams above (acting trivially on Mbl(Λ)).

2) The projective limit

S reg
Kp

(G, D) = lim←−Kp S

KpKp(G, D)

satisfies the “dvr extension property”: For every dvr R of mixed characteristic (0, p)

we have:

S reg
Kp

(G, D)(R) = ShKp(G, D)(R[1/p]).

Note that (a) and (b) together amount to the existence of a smooth morphism

¯qK : S reg

K (G, D) → [G\Mbl(Λ)]

where the target is the quotient algebraic stack.

Proof. By [10, Theorem 4.2.7], there are schemes SK (G, D) which satisfy similar properties,

excluding (c), but with Mbl(Λ) replaced by the PZ local model Mloc(Λ). In particular, we

have

(cid:102)SK (G, D)

qK

πK

(11.0.0.2)

Mloc(Λ)
with πK a G-torsor and qK smooth and G-equivariant. We set

SK (G, D)

(cid:102)S reg
K (G, D) = (cid:102)SK (G, D) ×

Mloc(Λ)

Mbl(Λ)

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which carries a diagonal G-action. Since r : Mbl(Λ) −→ Mloc(Λ) is given by a blow-up, is
projective, and we can see ([14, §2]) that the quotient

K : (cid:102)S reg

πreg

K (G, D) −→ S reg

K (G, D) := G\(cid:102)S reg

K (G, D)

is represented by a scheme and gives a G-torsor. (This is an example of a “linear modifica-
tion”, see [14, §2].) In fact, since blowing-up commutes with ´etale localization, S reg
K (G, D) is
the blow-up of SK (G, D) at the subscheme of closed points that correspond to ∗ ∈ Mloc(Λ)

under the local model diagram (11.0.0.1). This set of points is the discrete Kottwitz-
Rapoport stratum of the special fiber of SK (G, D). The projection gives a smooth G-

morphism

K : (cid:102)S reg

qreg

K (G, D) −→ Mbl(Λ)

which completes the local model diagram. Property (c) follows from Theorem 10.0.1 and

properties (a) and (b) which imply that S reg

K (G, D) and Mbl(Λ) are locally isomorphic for

the ´etale topology. The rest of the properties in the statement follow from the corresponding

properties for SK (G, D) and the construction. For a more detailed presentation of this

proof see the proof of [18, Theorem 7.2.1].

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63

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