# The Regularity of difference divisors

Baiqing Zhu

(Date: July 2023)

###### Abstract.

We prove the regularity of difference divisors on unitary and GSpin Rapoport-
Zink spaces.

###### Contents

  1. 1 Introduction
  2. 2 Special cycles on Rapoport-Zink spaces
  3. 3 Deformation theory
  4. 4 Regularity of the difference divisor

##  1\. Introduction

###  1.1. Background

For every positive integer $n\geq 1$, let $\mathcal{N}_{1,n-1}$ be the unitary
Rapoport-Zink space of signature $(1,n-1)$, which has been defined and
extensively studied in the works of Kudla and Rapoport [KR11], [KR14].
Terstiege introduced the difference divisors on the formal scheme
$\mathcal{N}_{1,n-1}$ in [Ter13a], he also proved the regularity of difference
divisors in [Ter13b]. A key step in the proof is a previous result in his
joint work with Rapoport and Zhang [RTZ13, Theorem 10.7], whose proof is based
on the windows theory developed by Zink in [Zin01] and [Zin02].

For a self-dual quadratic lattice $V$ of rank $m=n+1$, a Hodge type Rapoport-
Zink space $\mathcal{N}(V)$ has been constructed and studied in the works of
Kim [Kim18] and Howard-Pappas [HP17]. When $V$ is one of the self-dual
lattices of rank 4, Terstiege defined difference divisors on the formal scheme
$\mathcal{N}(V)\simeq\mathcal{M}^{HB}$ and studied the intersection numbers of
them in [Ter11], where he also proved the regularity of these difference
divisors. The notion of difference divisors can be defined similarly on
$\mathcal{N}(V)$ for general $V$, but the regularity of these difference
divisors was previously not known to us. It’s difficult to prove a similar
result as [RTZ13, Theorem 10.7] since the $p$-divisible groups parameterized
by the formal scheme $\mathcal{N}(V)$ have dimension $2^{n}$ and height
$2^{n+1}$, so the windows theory becomes much more complicated in this case.

###  1.2. Novelty of the method

In this article, we give a unified proof of the regularity of difference
divisors on $\mathcal{N}_{1,n-1}$ and $\mathcal{N}(V)$, see Theorem 4.2.4 and
its Corollary 4.2.5. In the unitary case, the proof is based on the
Grothendieck-Messing deformation theory of $p$-divisible groups of signature
$(1,n-1)$, see Lemma 3.4.2, which is simpler compared to Terstiege’s proof. In
the GSpin case, similar deformation results are also available in Madapusi’s
work [MP16, Proposition 5.16], see Lemma 3.4.3. Both Lemma 3.4.2 and Lemma
3.4.3 are of Grothendieck-Messing type, which makes computations easier. Our
main result is obtained by combining these two lemmas with a commutative
algebra result Lemma 4.2.2. We also identify the formally smooth locus of
difference divisors.

###  1.3. Applications

In the work of Li and Zhang [LZ22a, Lemma 2.8.1], they proved the linear
invariance of the derived intersections on the Rapoport-Zink space
$\mathcal{N}_{1,n-1}$ based on the regularity of difference divisors, which
guarantees that the local arithmetic intersection numbers on
$\mathcal{N}_{1,n-1}$ are well-defined.

The parallel result in the GSpin case is proved by Terstiege [Ter11, Lemma
4.1, Proposition 4.2] (see also [LZ22b, Lemma 4.11.1]) using globalization.
Our result can be applied to give an alternative proof of the linear
invariance of the derived intersections on the Rapoport-Zink space
$\mathcal{N}(V)$ along the lines of [LZ22a, Lemma 2.8.1] (see [LZ22b, foodnote
3]), which guarantees that the local arithmetic intersection numbers on
$\mathcal{N}(V)$ are well-defined.

Acknowledgements. The author is grateful to Professor Michael Rapoport and
Professor Chao Li for their careful reading of the original manuscript and
many helpful comments. The author is supported by the Department of
Mathematics at Columbia University in the city of New York.

##  2\. Special cycles on Rapoport-Zink spaces

###  2.1. Rapoport-Zink spaces

Fix an odd prime $p$. Let $F=\mathbb{Q}_{p}$, and
$\mathbb{F}=\overline{\mathbb{F}}_{p}$. Let $K$ be the completion of the
maximal unramified extension of $F$ with the Frobenius automorphism $\sigma$,
let $W$ be the integer ring of $K$. Let $\textup{Nilp}_{W}$ be the category of
$W$-schemes on which $p$ is locally nilpotent, for an object $S$ in
$\textup{Nilp}_{W}$, we use $\overline{S}$ to denote the scheme
$S\times_{W}\mathbb{F}$.

####  2.1.1. Unitary Rapoport-Zink space $\mathcal{N}^{u}=\mathcal{N}_{1,n-1}$

Let $E=\mathbb{Q}_{p^{2}}$ be the unramified quadratic extension of $F$, and
$\mathcal{O}_{E}$ be the integer ring of $E$. Let $\varpi=p$ be the
uniformizer of $\mathcal{O}_{F}$ and $\mathcal{O}_{E}$. Fix an $F$-algebra
embedding $\phi_{0}:\mathcal{O}_{E}\rightarrow W$ and denote by $\phi_{1}$ the
embedding $\sigma\circ\phi_{0}:\mathcal{O}_{E}\rightarrow W$. The embedding
$\phi_{0}$ induces an embedding between the residue fields
$\mathbb{F}_{p^{2}}\rightarrow\mathbb{F}$, which we shall think of as the
natural embedding. For any $\mathcal{O}_{E}$-module $\Lambda$, we shall write
$\Lambda_{W}$ for $\Lambda\otimes_{\mathcal{O}_{E},\phi_{0}}W$.

Let $n$ be a positive integer. For a $W$-scheme $S$, a unitary $p$-divisible
groups of signature $(1,n-1)$ over $S$ is a triple $(X,\iota,\lambda)$, where
(1) $X$ is a $p$-divisible group of dimension $n$ and height $2n$ over $S$;
(2) $\iota:\mathcal{O}_{E}\rightarrow\textup{End}(X)$ is an action satisfying
the signature $(1,n-1)$ condition, i.e., for $\alpha\in\mathcal{O}_{E}$,

$\textup{char}(\iota(\alpha):\textup{Lie}X)(T)=(T-\phi_{0}(\alpha))(T-\phi_{1}(\alpha))^{n-1}\in\mathcal{O}_{S}[T];$

(3) $\lambda:X\rightarrow X^{\vee}$ is a principal polarization such that the
associated Rosati involution induces $\alpha\mapsto\sigma(\alpha)$ on
$\mathcal{O}_{E}$ via $\iota$.

Over $\mathbb{F}$, there is a unique such triple
$\mathbb{X}=(\mathbb{X},\iota_{\mathbb{X}},\lambda_{\mathbb{X}})$ such that
$\mathbb{X}$ is supersingular, up to $\mathcal{O}_{E}$-linear isogeny
preserving the polarization up to scalar. The unitary Rapoport-Zink space of
signature $(1,n-1)$ is the following functor
$\mathcal{N}_{1,n-1}:\textup{Nilp}_{W}\rightarrow\textbf{Sets}$: for a scheme
$S$ in the category $\textup{Nilp}_{W}$, the set of $\mathcal{N}_{1,n-1}(S)$
is the isomorphism classes of quadruples $(X,\iota,\lambda,\rho)$, where
$(X,\iota,\lambda)$ is a unitary $p$-divisible group over $S$ of signature
$(1,n-1)$ and
$\rho:X\times_{S}\overline{S}\rightarrow\mathbb{X}\times_{\mathbb{F}}\overline{S}$
is an $\mathcal{O}_{E}$-linear quasi-isogeny of height zero which respects
$\lambda$ and $\lambda_{\mathbb{X}}$ up to a scalar
$c(\rho)\in\mathcal{O}_{F}^{\times}$ (i.e.,
$\rho^{\vee}\circ\lambda_{\mathbb{X}}\circ\rho=c(\rho)\cdot\lambda$).

######  Theorem 2.1.1.

The functor $\mathcal{N}_{1,n-1}$ is represented by a formal scheme which is
formally locally of finite type and formally smooth of relative dimension
$n-1$ over $\textup{Spf}\,W$.

###### Proof.

These facts about $\mathcal{N}_{1,n-1}$ are proved in [RZ96, Theorem 2.16],
[KR11, $\S$2.1] and [Vol10]. ∎

######  Remark 2.1.2.

For a general $p$-adic local field $F$, let $\mathcal{O}_{\breve{F}}$ be the
integer ring of the maximal unramified extension of $F$, the relative unitary
Rapoport-Zink space is constructed in Mihatsch’s work [Mih22], similar results
as Theorem 2.1.1 are also proved (replace $W$ by $\mathcal{O}_{\breve{F}}$,
see Proposition 2.17 of loc. cit.). Our method of proving the regularity of
difference divisors can be extended to the relative setting by [Mih22, Lemma
2.10] if we work with the divided power thickening
$(\mathcal{O}_{\breve{F}}/\varpi^{n+1}\rightarrow\mathcal{O}_{\breve{F}}/\varpi^{n})$
successively in proving Lemma 4 below.

In the following we denote $\mathcal{N}^{u}\coloneqq\mathcal{N}_{1,n-1}$, it
has relative dimension $n-1$ over $\textup{Spf}\,W$, let
$\mathbb{Y}=(\mathbb{Y},\iota_{\mathbb{Y}},\lambda_{\mathbb{Y}})$ be the
framing object for $n=1$, let
$(\overline{\mathbb{Y}},\iota_{\overline{\mathbb{Y}}},\lambda_{\overline{\mathbb{Y}}})=(\mathbb{Y},\iota_{\mathbb{Y}}\circ\sigma,\lambda_{\mathbb{Y}})$
be its conjugate.

####  2.1.2. GSpin Rapoport-Zink space $\mathcal{N}^{o}=\mathcal{N}(V)$

Let $k\geq 1$ be an integer, a quadratic lattice of rank $k$ is a pair
$(L,q_{L})$ such that $L$ is a free $\mathcal{O}_{F}$-module of rank $k$ and
$q_{L}:L\rightarrow F$ is a quadratic form on $L$. The quadratic form $q_{L}$
induces a symmetric bilinear form $(\cdot,\cdot):L\times L\rightarrow F$ by
$(x,y)=\frac{1}{2}(q(x+y)-q(x)-q(y))$. Let $L^{\sharp}=\\{x\in
L\otimes_{\mathcal{O}_{F}}F:(x,L)\subset\mathcal{O}_{F}\\}$. We say a
quadratic lattice is integral if $q_{L}(x)\in\mathcal{O}_{F}$ for all $x\in
L$, is self-dual if it is integral and $L$ = $L^{\sharp}$.

Let $n\geq 2$ be an integer, and $V$ be a self-dual quadratic lattice of rank
$m=n+1$, associated to $V$ we have a local unramified Shimura Hodge data
$(G,b,\mu,C)$ (in the sense of [HP17, Definition 2.2.4]) constructed in [HP17,
Proposition 4.2.6], where $G=\textup{GSpin}(V)$, $b\in G(K)$ is a basic
element, $\mu:\mathbb{G}_{m}\rightarrow G$ is a certain cocharacter, and
$C=C(V)$ is the Clifford algebra of $V$. Let
$C^{\vee}=\textup{Hom}_{\mathcal{O}_{F}}(C,\mathcal{O}_{F})$ be the linear
dual of $C$. By [HP17, Lemma 2.2.5], this local unramified Shimura-Hodge data
gives rise to a (unique up to isomorphism) supersingular $p$-divisible group
$\mathbb{X}_{V}$ over $\mathbb{F}$ whose contravariant Dieudonne module
$\mathbb{D}(\mathbb{X}_{V})(W)$ is given by $C^{\vee}_{W}\coloneqq
C^{\vee}\otimes_{\mathcal{O}_{F}}W$ with Frobenius $\mathbf{F}=b\circ\sigma$,
it is also equipped with a $p$-principal polarization
$\lambda_{0}:\mathbb{X}_{V}\rightarrow\mathbb{X}_{V}^{\vee}$. Associated to
the local unramified Shimura-Hodge data $(G,b,\mu,C)$, we have a GSpin
Rapoport–Zink space $\textup{RZ}(V)=\textup{RZ}(G,b,\mu,C)$ of Hodge type
([HP17, $\S$4.3], see also [Kim18]) parametrizing $p$-divisible groups with
crystalline Tate tensors.

######  Theorem 2.1.3.

The GSpin Rapoport-Zink space $\textup{RZ}(V)$ is formally locally of finite
type and formally smooth of relative dimension $n-1$ over $\textup{Spf}\,W$.

###### Proof.

This is Theorem B in [HP17]. ∎

Let $X^{\textup{univ}}$ be the universal $p$-divisible group over
$\textup{RZ}(V)$ with the universal quasi-isogeny
$\rho^{\textup{univ}}:\mathbb{X}_{V}\times_{\mathbb{F}}\overline{\textup{RZ}(V)}\rightarrow
X^{\textup{univ}}\times_{\textup{RZ}(V)}\overline{\textup{RZ}(V)}$. Let
$\mathcal{N}^{o}=\mathcal{N}(V)$ be the connected component of
$\textup{RZ}(V)$ such that the $p$-principal polarization $\lambda_{0}$ of
$\mathbb{X}_{V}$ lifts to a $p$-principal polarization
$\lambda^{\textup{univ}}$ on $X^{\textup{univ}}$ via the universal quasi-
isogeny $\rho^{\textup{univ}}$ (cf. [LZ22b, $\S$4.5]).

###  2.2. The space of special quasi-homorphisms

####  2.2.1. The unitary case

The space of special homomorphisms is the $\mathcal{O}_{E}$-module
$\textup{Hom}_{\mathcal{O}_{E}}(\overline{\mathbb{Y}},\mathbb{X})$. There is a
natural $\mathcal{O}_{E}$-valued $\sigma$-Hermitian form on
$\textup{Hom}_{\mathcal{O}_{E}}(\overline{\mathbb{Y}},\mathbb{X})$ given by

$(x,y)\mapsto\lambda_{\overline{\mathbb{Y}}}^{-1}\circ
y^{\vee}\circ\lambda_{\mathbb{X}}\circ
x\in\textup{End}_{\mathcal{O}_{E}}(\overline{\mathbb{Y}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{E}.$

In the following we denote
$\mathbb{V}^{u}=\textup{Hom}_{\mathcal{O}_{E}}^{\circ}(\overline{\mathbb{Y}},\mathbb{X})\coloneqq\textup{Hom}_{\mathcal{O}_{E}}(\overline{\mathbb{Y}},\mathbb{X})[\frac{1}{\varpi}]$.

####  2.2.2. The GSpin case

The inclusion $V\subset C^{\textup{op}}$ realizes
$V\subset\textup{End}_{W}(C^{\vee}_{W})$ as special homomorphisms of
$C^{\vee}_{W}$. Tensoring with $K$ gives a subspace
$V_{K}\subset\textup{End}_{K}(C^{\vee}_{K})$. Define the $\sigma$-linear
operator $\Phi=\overline{b}\circ\sigma$ on $V_{K}$, where
$\overline{b}\in\textup{SO}(V)(K)$ is the image of $b\in G(K)$ under the
natural quotient map $G=\textup{GSpin}(V)\rightarrow\textup{SO}(V)$. Then
$(V_{K},\Phi)$ is an isocrystal. The $\Phi$-fixed vectors form an $F$-vector
subspace of dimension $m=n+1$,

$\mathbb{V}^{o}\coloneqq
V_{K}^{\Phi}\subset\textup{End}^{\circ}(\mathbb{X})\coloneqq\textup{End}(\mathbb{X})[1/\varpi],$

this space is called the space of special quasi-homomorphisms of
$\mathbb{X}_{V}$. The restriction of the quadratic form to $\mathbb{V}^{o}$
satisfies $x\circ x=q_{\mathbb{V}^{o}}(x)\cdot\textup{id}_{\mathbb{X}}$ for
$x\in\mathbb{V}^{o}$ where $q_{\mathbb{V}^{o}}$ is the quadratic form on
$\mathbb{V}^{o}$ defined by the base change of the quadratic form $q_{V}$ on
$V$.

###  2.3. Special cycles and difference divisors

######  Definition 2.3.1.

For any subset $L\subset\mathbb{V}^{u}$, define the special cycle
$\mathcal{Z}(L)$ in $\mathcal{N}^{u}$ as the following subfunctor: for any
object $S$ in $\textup{Nilp}_{W}$, the set $\mathcal{Z}(L)(S)$ consists of
elements $(X,\iota_{X},\lambda_{X},\rho)\in\mathcal{N}^{u}(S)$ such that the
quasi-homomorphism $\rho^{-1}\circ
x\circ\rho_{\overline{Y}}:\overline{Y}\times_{S}\overline{S}\rightarrow
X\times_{S}\overline{S}$ extends to a homomorphism from $\overline{Y}$ to $X$
for all $x\in L$.

For any subset $L\subset\mathbb{V}^{o}$, define the special cycle
$\mathcal{Z}(L)\subset\mathcal{N}^{o}$ to be the closed formal subscheme cut
out by the condition

$\rho^{\textup{univ}}\circ
x\circ(\rho^{\textup{univ}})^{-1}\subset\textup{End}(X^{\textup{univ}}),$

for all $x\in L$.

######  Remark 2.3.2.

We use the same symbol $\mathcal{Z}(L)$ for special cycles, although they may
lie in different formal schemes. In the following discussion, we will always
make clear on which space the special cycle lie in.

######  Proposition 2.3.3.

The special cycle functor $\mathcal{Z}(L)$ is represented by a closed formal
subscheme of $\mathcal{N}^{u}$ (resp. $\mathcal{N}^{o}$). In fact, for any
$x\in\mathbb{V}^{u}$ ($x\in\mathbb{V}^{o}$) such that
$(x,x)\in\mathcal{O}_{E}\backslash\\{0\\}$ (resp.
$q_{\mathbb{V}^{o}}(x)\in\mathcal{O}_{F}\backslash\\{0\\}$), the special cycle
$\mathcal{Z}(x)$ is an effective Cartier divisor in $\mathcal{N}^{u}$ (resp.
$\mathcal{N}^{o}$) and flat over $W$.

###### Proof.

For the unitary case, this is proved in [KR11, Proposition 3.5, Lemma 3.7].
For the GSpin case, this is proved in [LZ22b, Proposition 4.10.1] ∎

For a point $z\in\mathcal{N}^{u}(\mathbb{F})$ (resp.
$\mathcal{N}^{o}(\mathbb{F})$), let $\mathcal{O}_{\mathcal{N}^{u},z}$ (resp.
$\mathcal{O}_{\mathcal{N}^{o},z}$) be the completed local ring of
$\mathcal{N}^{u}$ (resp. $\mathcal{N}^{o}$) at $z$, then Theorem 2.1.1 (resp.
Theorem 2.1.3) implies that $\mathcal{O}_{\mathcal{N}^{u},z}$ (resp.
$\mathcal{O}_{\mathcal{N}^{o},z}$) $\simeq
W[[t_{1},t_{2},\cdot\cdot\cdot,t_{n-1}]]$. For a special quasi-homomorphism
$x$, the completed local ring $\mathcal{O}_{\mathcal{Z}(x),z}$ of
$\mathcal{Z}(x)$ at $z$ is cut out by a single equation $f_{x,z}\in
W[[t_{1},t_{2},\cdot\cdot\cdot,t_{n-1}]]$ such that $\varpi\nmid f_{x,z}$,
i.e., $\mathcal{O}_{\mathcal{Z}(x),z}\simeq
W[[t_{1},t_{2},\cdot\cdot\cdot,t_{n-1}]]/(f_{x,z})$. If
$z\notin\mathcal{Z}(x)(\mathbb{F})$, we can take $f_{x,z}=1$, otherwise
$f_{x,z}$ belongs to the maximal ideal
$\mathfrak{m}_{z}=(\varpi,t_{1},\cdots,t_{n-1})$ and is determined up to some
units in the ring $W[[t_{1},t_{2},\cdot\cdot\cdot,t_{n-1}]]$.

For an element $x\in\mathbb{V}^{u}$ (resp. $x\in\mathbb{V}^{o}$) such that
$(x,x)\in\mathcal{O}_{E}\backslash\\{0\\}$ (resp.
$q_{\mathbb{V}^{o}}(x)\in\mathcal{O}_{F}\backslash\\{0\\}$), we have a natural
closed immersion of closed formal schemes
$\mathcal{Z}(\varpi^{-1}x)\hookrightarrow\mathcal{Z}(x)$ by Definition 2.3.1.
The closed immersion $\mathcal{Z}(\varpi^{-1}x)\hookrightarrow\mathcal{Z}(x)$
impies the divisibility $f_{\varpi^{-1}x,z}|f_{x,z}$ at every point
$z\in\mathcal{N}^{u}(\mathbb{F})$ (resp. $\mathcal{N}^{o}(\mathbb{F})$).

######  Definition 2.3.4.

Let $x\in\mathbb{V}^{u}$ (resp. $x\in\mathbb{V}^{o}$) be an element such that
$(x,x)\in\mathcal{O}_{E}\backslash\\{0\\}$ (resp.
$q_{\mathbb{V}^{o}}(x)\in\mathcal{O}_{F}\backslash\\{0\\}$). Define the
difference divisor associated to $x$ to be the following Cartier divisor on
$\mathcal{N}^{u}$ (resp. $\mathcal{N}^{o}$),

$\mathcal{D}(x)\coloneqq\mathcal{Z}(x)-\mathcal{Z}(\varpi^{-1}x).$

i.e., at a point $z\in\mathcal{Z}(x)(\mathbb{F})$, suppose $\mathcal{Z}(x)$ is
cut out by $f_{x,z}\in\mathcal{O}_{\mathcal{N}^{u},z}$ and
$\mathcal{Z}(\varpi^{-1}x)$ is cut out by
$f_{\varpi^{-1}x,z}\in\mathcal{O}_{\mathcal{N}^{u},z}$ (resp.
$\mathcal{O}_{\mathcal{N}^{o},z}$), then $\mathcal{D}(x)$ is cut out by
$d_{x,z}\coloneqq
f_{x,z}/f_{\varpi^{-1}x,z}\in\mathcal{O}_{\mathcal{N}^{u},z}$ (resp.
$\mathcal{O}_{\mathcal{N}^{o},z}$).

##  3\. Deformation theory

###  3.1. Preliminaries on linear algebra

######  Definition 3.1.1.

Let $R$ be a commutative ring, let $D$ be a free $R$-module of finite rank
$n$, a hyperplane $P$ in $D$ is a direct summand of $D$ which is free of rank
$n-1$ over $R$, a line $L$ in $D$ is a direct summand of $D$ which is free of
rank $1$ over $R$.

######  Lemma 3.1.2.

Let $R$ be a commutative ring, let $D$ be a free $R$-module of finite rank
$n$, then we have a bijection,

$\displaystyle K:\\{\textup{Lines
in}\,\,D^{\vee}\coloneqq\textup{Hom}_{R}(D,R).\\}$
$\displaystyle\longrightarrow\\{\textup{Hyperplanes in $D$}.\\},$
$\displaystyle L=R\cdot l$ $\displaystyle\longmapsto P=\textup{ker}(l).$

###### Proof.

This can be proved by purely linear algebra. ∎

###  3.2. The Dieudonné module of an $\mathbb{F}$-point of $\mathcal{N}^{u}$

A point $z\in\mathcal{N}^{u}(\mathbb{F})$ corresponds to a $p$-divisible group
$X$ of signature $(1,n-1)$ over $\mathbb{F}$ which is isogenous to the basic
framing object $\mathbb{X}$, the dimension of $\mathbb{X}$ is $n$ and the
height of $\mathbb{X}$ is $2n$, hence the Dieudonné module $\mathbb{D}(X)$ of
$X$ is a free $W$-module of rank $2n$. Let $D\coloneqq\mathbb{D}(X)$, it is
equipped with the following exact sequence,

$0\longrightarrow\textbf{F}^{1}D_{\mathbb{F}}\longrightarrow
D_{\mathbb{F}}\longrightarrow\textup{Lie}(X)\longrightarrow 0.$

where both $\textbf{F}^{1}D_{\mathbb{F}}$ and $\textup{Lie}(X)$ are
$\mathbb{F}$-vector spaces of dimension $n$.

The $W$-module module $D$ admits $\mathcal{O}_{E}\otimes W\simeq W\times W$
action through $\iota:\mathcal{O}_{E}\rightarrow\textup{End}(X)$ and
decomposes into

$D=\mathbb{D}_{0}(X)\oplus\mathbb{D}_{1}(X),$

where both $\mathbb{D}_{0}(X)$ and $\mathbb{D}_{1}(X)$ are free $W$-modules of
rank $n$, and $\mathcal{O}_{E}$ acts on $\mathbb{D}_{0}(X)$ (resp.
$\mathbb{D}_{1}(X)$) by multiplication through
$\phi_{0}:\mathcal{O}_{E}\rightarrow W$ (resp.
$\phi_{1}:\mathcal{O}_{E}\rightarrow W$). For simplicity, let
$\mathbf{D}\coloneqq\mathbb{D}_{0}(X)$ and
$\overline{\mathbf{D}}\coloneqq\mathbb{D}_{1}(X)$. The signature $(1,n-1)$
condition implies that the $\mathbb{F}$-vector space
$\textbf{F}^{1}D_{\mathbb{F}}$ decomposes as

$\textbf{F}^{1}D_{\mathbb{F}}=\textbf{F}^{1}D_{\mathbb{F}}\cap\mathbf{D}_{\mathbb{F}}\oplus\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}.$

where
$\textup{dim}_{\mathbb{F}}\,\textbf{F}^{1}D_{\mathbb{F}}\cap\mathbf{D}_{\mathbb{F}}=n-1$
and
$\textup{dim}_{\mathbb{F}}\,\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}=1$.

The $p$-principal polarization $\lambda:X\rightarrow X^{\vee}$ induces a non-
degenerate symplectic form $\langle\cdot,\cdot\rangle$ on the rank $2n$ free
$W$-module $\mathbb{D}(X)$, the compatibility with $\mathcal{O}_{E}$-action
implies that both $\mathbf{D}$ and $\overline{\mathbf{D}}$ are totally
isotropic subspaces of $\mathbb{D}(X)$ under this symplectic form, hence there
is a prefect pairing

$\mathbf{D}\times\overline{\mathbf{D}}\stackrel{{\scriptstyle\langle\cdot,\cdot\rangle}}{{\longrightarrow}}W,$

i.e., we have isomorphisms
$\mathbf{D}\simeq\overline{\mathbf{D}}^{\vee}\coloneqq\textup{Hom}_{W}(\overline{\mathbf{D}},W)$
and
$\overline{\mathbf{D}}\simeq\mathbf{D}^{\vee}\coloneqq\textup{Hom}_{W}(\mathbf{D},W)$,
both two isomorphisms are induced by the symplectic form
$\langle\cdot,\cdot\rangle$. The pairing
$\langle\cdot,\cdot\rangle_{\mathbb{F}}:\mathbf{D}_{\mathbb{F}}\times\overline{\mathbf{D}}_{\mathbb{F}}\rightarrow\mathbb{F}$
is also perfect. Moreover, the hyperplane
$\textbf{F}^{1}D_{\mathbb{F}}\cap\mathbf{D}_{\mathbb{F}}\subset\mathbf{D}_{\mathbb{F}}$
and the line
$\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}\subset\overline{\mathbf{D}}_{\mathbb{F}}$
are annihilators of each other under this pairing.

The Frobenius action $\Phi$ on $\mathbb{D}(X)$ is degree $1$, i.e.,
$\Phi(\mathbf{D})\subset\overline{\mathbf{D}}$ and
$\Phi(\overline{\mathbf{D}})\subset\mathbf{D}$, hence
$\Phi^{2}(\mathbf{D})\subset\mathbf{D}$. The isocyrstal
$(\mathbf{D}[\frac{1}{\varpi}],\Phi)$ is basic since the framing object
$\mathbb{X}$ is supersingular, let $C$ be the $\varpi^{-1}\Phi^{2}$-invariant
elements in the $K$-space $\mathbf{D}[\frac{1}{\varpi}]$, i.e.,
$C=(\mathbf{D}[\frac{1}{\varpi}])^{\varpi^{-1}\Phi^{2}}$, it is a
$n$-dimensional $E$-vector space. Fix $\delta\in\mathcal{O}_{E}^{\times}$ such
that $\sigma(\delta)=-\delta$, define a non-degenerate $\sigma$-Hermitian form
on $C$ by

$\\{x,y\\}\coloneqq(\varpi\delta)^{-1}\langle x,\Phi y\rangle.$

then we have an isomorphism of $\sigma$-Hermitian spaces by [KR11, Lemma 3.9]
(see also [LZ17, $\S$2.3]),

$i_{\textup{crys},z}:\mathbb{V}^{u}=\textup{Hom}_{\mathcal{O}_{E}}^{\circ}(\overline{\mathbb{Y}},\mathbb{X})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}C.$

Therefore we may view elements of $C$ as special quasi-homomorphisms. If
$z\in\mathcal{Z}(x)(\mathbb{F})$ for some $x\in\mathbb{V}^{u}$, then
$x_{\textup{crys},z}\coloneqq i_{\textup{crys},z}(x)\in C$. In fact, we have
$x_{\textup{crys},z}\in C\cap\mathbf{D}$, and its image
$\overline{x_{\textup{crys},z}}$ in $\mathbf{D}_{\mathbb{F}}$ is contained in
the hyperplane $\textbf{F}^{1}D_{\mathbb{F}}\cap\mathbf{D}_{\mathbb{F}}$,
i.e., orthogonal to the line
$\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}$ under the
pairing $\langle\cdot,\cdot\rangle_{\mathbb{F}}$.

###  3.3. The Dieudonné module of an $\mathbb{F}$-point of $\mathcal{N}^{o}$

Let $S$ be an affine $W$-scheme such that $\mathcal{O}_{S}$ is a noetherian
$\varpi$-adically complete $W$-algebra, for any point
$z\in\mathcal{N}^{o}(S)$, there is an
$\mathcal{O}_{S/W}^{\textup{crys}}$-module crystal
$\mathbf{V}_{\textup{crys},z}$ of rank $m$ constructed in [MP16, $\S$4.5,
$\S$4.6] satisfying that
$\bullet$ The projective $S$-module $\mathbf{V}_{\textup{crys},z}(S)$ contains
a canonical isotropic line $\textup{Fil}^{1}\mathbf{V}_{\textup{crys},z}(S)$.
$\bullet$ For any surjection $R\rightarrow S$ in $\textup{Alg}_{W}$ whose
kernel admits divided powers, we have $\mathbf{V}_{\textup{crys},z}(R)$ a
projective $R$-module of rank $m$. It is equipped with a non-degenerate
symmetric $R$-bilinear form.
$\bullet$ Suppose $z\in\mathcal{Z}(x)(S)$ for some $x\in\mathbb{V}^{o}$, then
the crystalline realization $x_{\textup{crys},z}$ of the special quasi-
homomorphism $x$ belongs to $\mathbf{V}_{\textup{crys},z}(S)$ and orthogonal
to the line $\textup{Fil}^{1}\mathbf{V}_{\textup{crys},z}(S)$.
$\bullet$ Let $S=\textup{Spec}\,\mathbb{F}$. The evaluation
$\mathbf{V}\coloneqq\mathbf{V}_{\textup{crys},z}(W)$ at the divided power
thickening morphism
$\textup{Spec}\,\mathbb{F}\hookrightarrow\textup{Spec}\,W$, viewed as a
$W$-module, has the structure of a self-dual $W$-quadratic lattice which can
also be viewed as a sub-lattice of $V_{K}$, hence has the Frobenius
$\Phi$-action.

####  3.3.1. Strong divisibility

Let $z\in\mathcal{N}^{o}(\mathbb{F})$ be a point, let
$z^{\prime}\in\mathcal{N}^{o}(W)$ be a lift of $z$ to $W$. The $W$-module
$\mathbf{V}=\mathbf{V}_{\textup{crys},z}(W)$ is a quadratic $W$-lattice
containing an isotropic line
$L^{1}=\textup{Fil}^{1}\mathbf{V}_{\textup{crys},z}(W)$. Let
$L^{0}\coloneqq(L^{1})^{\bot}\subset\mathbf{V}$, we have a filtration
$0\subset L^{0}\subset L^{1}\subset\mathbf{V}$, the following lemma proves
that this filtration is strongly divisible.

######  Lemma 3.3.1.

The $\Phi$-action on $V_{K}$ induces the following Frobenius-linear
isomorphism of $W$-lattices,

$\Phi:\varpi^{-1}L^{1}+L^{0}+\varpi\mathbf{V}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbf{V}$

###### Proof.

This is proved in [MP16, $\S$4.8]. ∎

######  Lemma 3.3.2.

Let $\nu_{\varpi}:K\rightarrow\mathbb{Z}\cup\\{\infty\\}$ be the $\varpi$-adic
valuation of $K$. Let $(\cdot,\cdot)$ be the bilinear pairing induced by the
self-dual quadratic form $q_{\mathbf{V}}$ on $\mathbf{V}$, let
$l\in\mathbf{V}$ be a generator of the isotropic line $L^{1}$, then

$L^{0}+\varpi\mathbf{V}=\\{x\in\mathbf{V}:\nu_{\varpi}((x,l))\geq 1\\}.$

###### Proof.

By the definition of $L^{0}$, we have $\nu_{\varpi}((x,l))\geq 1$ for any
$x\in L^{0}+\varpi\mathbf{V}$. Now if $x\in\mathbf{V}$ satisfies the condition
that $\nu_{\varpi}((x,l))\geq 1$, there exists $x^{\prime}\in\mathbf{V}$ such
that $(l,x^{\prime})=\varpi^{-1}(l,x)$ since $l\notin\varpi\mathbf{V}$ and
$\mathbf{V}$ is a self-dual quadratic lattice, therefore $x-\varpi\cdot
x^{\prime}\in L^{0}$, hence $x\in L^{0}+\varpi\mathbf{V}$. ∎

###  3.4. Deformation theory

######  Definition 3.4.1.

Let $\mathscr{C}$ be the following category:
$\bullet$ Objects in $\mathscr{C}$ are triples
$(\mathcal{O},\mathcal{O}\rightarrow\mathbb{F},\delta)$, where $\mathcal{O}$
is a local Artinian $W$-algebra, $\mathcal{O}\rightarrow\mathbb{F}$ is a
$W$-algebra map, and $\delta$ is a nilpotent divided power structure on
$\textup{ker}(\mathcal{O}\rightarrow\mathbb{F})$ (cf. [BO78, Definitions 3.1,
3.27]).
$\bullet$ Morphisms in $\mathscr{C}$ are $W$-algebra maps that are compatible
with the structure maps to $\mathbb{F}$ and the divided power structures.

####  3.4.1. The unitary case

For an $\mathbb{F}$-point $z\in\mathcal{N}^{u}(\mathbb{F})$ which corresponds
to a $p$-divisible group $X$, let $D=\mathbb{D}(X)$ be the Dieudonné module of
$\mathbb{X}$, let $\mathbf{D}\coloneqq\mathbb{D}_{0}(X)$ and
$\overline{\mathbf{D}}\coloneqq\mathbb{D}_{1}(X)$ be the two sublattices of
$D$ as in $\S$3.2. Let $\hat{\mathcal{N}}^{u}_{z}$ be the completion of the
formal scheme $\mathcal{N}^{u}$ at $z$. Let $\mathcal{O}\in\mathscr{C}$, an
element $\tilde{z}\in\hat{\mathcal{N}}^{u}_{z}(\mathcal{O})$ corresponds to a
$p$-divisible group of signature $(1,n-1)$ over $\mathcal{O}$ deforming that
over $\mathbb{F}$ defined by $z$. Let
$D_{\mathcal{O}}=D\otimes_{W}\mathcal{O}$, the $p$-divisible group
corresponding to $\tilde{z}$ gives rise to a filtration
$\mathbf{F}_{\tilde{z}}^{1}D_{\mathcal{O}}\subset D_{\mathcal{O}}$ by
Grothendieck-Messing theory. Define $f_{\mathcal{O}}(\tilde{z})$ to be the
intersection
$\mathbf{F}_{\tilde{z}}^{1}D_{\mathcal{O}}\cap\mathbf{D}_{\mathcal{O}}$ inside
$D_{\mathcal{O}}$. By the signature $(1,n-1)$ condition,
$f_{\mathcal{O}}(\tilde{z})$ is a hyperplane in $\mathbf{D}_{\mathcal{O}}$. It
also lifts $\textbf{F}^{1}D_{\mathbb{F}}\cap\mathbf{D}_{\mathbb{F}}$ by
construction. Thus we have defined a map

$f_{\mathcal{O}}:\hat{\mathcal{N}}^{u}_{z}(\mathcal{O})\rightarrow\\{\textup{Hyperplanes
in $\mathbf{D}_{\mathcal{O}}$ lifting
$\textbf{F}^{1}D_{\mathbb{F}}\cap\mathbf{D}_{\mathbb{F}}$.}\\}.$

By Lemma 3.1.2, we have the following bijection

$\displaystyle\\{\textup{Lines in $\overline{\mathbf{D}}_{\mathcal{O}}$}\,\,$
$\displaystyle\textup{lifting the line
$\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}\subset\overline{\mathbf{D}}_{\mathbb{F}}$}.\\}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}$
$\displaystyle\\{\textup{Hyperplanes in $\mathbf{D}_{\mathcal{O}}$ lifting the
hyperplane
$\textbf{F}^{1}D_{\mathbb{F}}\cap\mathbf{D}_{\mathbb{F}}\subset\mathbf{D}_{\mathbb{F}}$.}\\}$

######  Lemma 3.4.2.

Let $\mathcal{O}\in\mathscr{C}$, let $z\in\mathcal{N}^{u}(\mathbb{F})$ be an
$\mathbb{F}$-point, then there is a natural bijection

$\left\\{\textup{Lifts}\,\,z^{\prime}\in\widehat{\mathcal{N}}^{u}_{z}(\mathcal{O})\,\,\textup{of}\,\,z.\right\\}\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\left\\{\textup{Lines
in $\overline{\mathbf{D}}_{\mathcal{O}}$}\,\,\textup{lifting the line
$\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}$}\right\\}.$

Let $L\subset\mathbb{V}^{u}$ be an $\mathcal{O}_{F}$-lattice of rank $r\geq
1$. Let $z\in\mathcal{Z}(L)(\mathbb{F})$ and $\widehat{\mathcal{Z}(L)}_{z}$ be
the completion of $\mathcal{Z}(L)$ at $z$, then there is a natural bijection

$\left\\{\textup{Lifts}\,\,z^{\prime}\in\widehat{\mathcal{Z}(L)}_{z}(\mathcal{O})\,\,\textup{of}\,\,z.\right\\}\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\left\\{\begin{array}[]{ll}\textup{Lines
in $\overline{\mathbf{D}}_{\mathcal{O}}$ lifting
$\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}$}\\\
\textup{and orthogonal to $x_{\textup{crys},z}$ for any $x\in
L$.}\end{array}\right\\}.$

###### Proof.

This is proved by combining [LZ17, Theorem 3.1.3] and Lemma 3.1.2. ∎

####  3.4.2. The GSpin case

Let $z\in\mathcal{N}^{o}(\mathbb{F})$ be an $\mathbb{F}$-point, we use
$\mathbf{V}$ to denote the $\mathcal{O}_{\mathbb{F}/W}^{\textup{crys}}$-module
constructed in $\S$3.3, it is equipped with a filtration
$0\subset\textup{Fil}^{1}\mathbf{V}\subset\mathbf{V}$. Let
$\widehat{\mathcal{N}}_{z}$ be the completion of $\mathcal{N}$ at $z$. For an
object $\mathcal{O}\in\mathscr{C}$, let
$\mathbf{V}_{\mathcal{O}}=\mathbf{V}\otimes_{W}\mathcal{O}$.

######  Lemma 3.4.3.

Let $\mathcal{O}\in\mathscr{C}$, let $z\in\mathcal{N}^{o}(\mathbb{F})$ be an
$\mathbb{F}$-point, then there is a natural bijection

$\left\\{\textup{Lifts}\,\,z^{\prime}\in\widehat{\mathcal{N}}^{o}_{z}(\mathcal{O})\,\,\textup{of}\,\,z.\right\\}\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\left\\{\textup{Isotropic
$\mathcal{O}$-lines
in}\,\,\mathbf{V}_{\mathcal{O}}\,\,\textup{lifting}\,\,\textup{Fil}^{1}\mathbf{V}(\mathbb{F}).\right\\}.$

Let $L\subset\mathbb{V}^{o}$ be a $\mathbb{Z}_{p}$-lattice of rank $r\geq 1$.
Let $z\in\mathcal{Z}(L)(\mathbb{F})$ and $\widehat{\mathcal{Z}(L)}_{z}$ be the
completion of $\mathcal{Z}(L)$ at $z$, then there is a natural bijection

$\left\\{\textup{Lifts}\,\,z^{\prime}\in\widehat{\mathcal{Z}(L)}_{z}(\mathcal{O})\,\,\textup{of}\,\,z.\right\\}\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\left\\{\begin{array}[]{ll}\textup{Isotropic
$\mathcal{O}$-lines in $\mathbf{V}_{\mathcal{O}}$ lifting
$\textup{Fil}^{1}\mathbf{V}(\mathbb{F})$}\\\ \textup{and orthogonal to
$x_{\textup{crys},z}$ for any $x\in L$.}\end{array}\right\\}.$

###### Proof.

This is proved in [MP16, Proposition 5.16] (see also [LZ22b, Lemma 4.6.2]). ∎

##  4\. Regularity of the difference divisor

###  4.1. Formally smooth locus of the special cycle $\mathcal{Z}(L)$

######  Lemma 4.1.1.

Let $n\geq 2$ be an integer, let
$R=\mathcal{O}[[t_{1},\cdot\cdot\cdot,t_{n-1}]]$ where $\mathcal{O}$ is a
discrete valuation ring of characteristic $(0,p)$ with uniformizer $\pi$ and
residue field $\mathbf{k}$, the ring $R$ has maximal ideal
$\mathfrak{m}_{R}\coloneqq(\pi,t_{1},\cdot\cdot\cdot,t_{n-1})$. Let
$f_{1},f_{2},\cdots,f_{r}$ be $r$ elements in $\mathfrak{m}_{R}$, then the
quotient ring $\overline{R}=R/(f_{1},\cdots,f_{r})$ is formally smooth over
$\mathcal{O}$ of relative dimension $n-r-1$ if and only if its base change
$\overline{R}_{\mathbf{k}}=\overline{R}\otimes_{\mathcal{O}}\mathbf{k}$ is
formally smooth over $\mathbf{k}$ of relative dimension $n-r-1$.

###### Proof.

Let $\mathbf{0}:R\rightarrow\mathcal{O}$ be the continuous homomorphism which
sends all the $t_{j}$ to $0$. Let $J=(\frac{\partial f_{i}}{\partial
t_{j}}(\mathbf{0}))_{\begin{subarray}{c}1\leq i\leq r\\\ 1\leq j\leq
n-1\end{subarray}}$ be the Jacobian matrix of $f_{1},f_{2},\cdots,f_{r}$ at
$\mathbf{0}$. The quotient ring $\overline{R}$ is formally smooth over
$\mathcal{O}$ of relative dimension $n-r-1$ if and only if the matrix $J$ has
a $r\times r$-minor which is invertible in $\mathcal{O}$, because in this case
we can rearrange a system of parameters
$t_{1}^{\prime},\cdots,t_{n-1}^{\prime}$ of $R$ such that
$t_{i}^{\prime}=f_{i}$ for $1\leq i\leq r$.

Let $J_{\mathbf{k}}=J\otimes_{\mathcal{O}}\mathbf{k}$ be the base change of
$J$ to $\mathbf{k}$, the matrix $J$ has a $r\times r$-minor which is
invertible in $\mathcal{O}$ is equivalent to the rank of $J_{\mathbf{k}}$ is
$r$, which is further equivalent to $\overline{R}_{\mathbf{k}}$ is formally
smooth over $\mathbf{k}$ of relative dimension $n-r-1$, hence the quotient
ring $\overline{R}=R/(f_{1},\cdots,f_{r})$ is formally smooth over
$\mathcal{O}$ of relative dimension $n-r-1$ if and only if
$\overline{R}_{\mathbf{k}}$ is formally smooth over $\mathbf{k}$ of relative
dimension $n-r-1$. ∎

####  4.1.1. The unitary case

For a point $z\in\mathcal{N}^{u}(\mathbb{F})$, let
$\widehat{\mathcal{N}}^{u}_{z}$ be the completion of the formal scheme
$\mathcal{N}^{u}$ at $z$. Let $D=\mathbb{D}(X)$ be the Dieudonné module of
$\mathbb{X}$, let $\mathbf{D}\coloneqq\mathbb{D}_{0}(X)$ and
$\overline{\mathbf{D}}\coloneqq\mathbb{D}_{1}(X)$ be the two sublattices of
$D$ as in $\S$3.2. Let $L\subset\mathbb{V}^{u}$ be a $\mathcal{O}_{E}$-lattice
and suppose that $z\in\mathcal{Z}(L)(\mathbb{F})$, then we have an isometric
map $i_{\textup{crys},z}:L_{W}\coloneqq
L\otimes_{\mathcal{O}_{E}}W\rightarrow\mathbf{D}$ by extending $W$-linearly
the map
$i_{\textup{crys},z}:\mathbb{V}^{u}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}C\subset\mathbf{D}[\frac{1}{\varpi}]$
and restricting to $L_{W}$. We say the isometric map $i_{\textup{crys},z}$ is
primitive at $z$ if the induced map
$\overline{i_{\textup{crys},z}}:L_{\mathbb{F}}\rightarrow\mathbf{D}_{\mathbb{F}}$
is injective.

######  Lemma 4.1.2.

Let $L\subset\mathbb{V}^{u}$ be an $\mathcal{O}_{E}$-lattice of rank $1\leq
r\leq n-1$ and $z\in\mathcal{Z}(L)(\mathbb{F})$, then $\mathcal{Z}(L)$ is
formally smooth over $W$ of relative dimension $n-r-1$ at $z$ if and only if
the isometric map $i_{\textup{crys},z}$ is primitive at $z$.

###### Proof.

The special cycle $\mathcal{Z}(L)$ is cut out by $r$ equations in the
deformation space
$\widehat{\mathcal{N}}^{u}_{z}\simeq\textup{Spf}\,\mathcal{O}_{\mathcal{N}^{u},z}$
by Proposition 2.3.3, therefore the special cycle $\mathcal{Z}(L)$ is formally
smooth over $W$ of relative dimension $n-r-1$ at $z$ if and only if
$\mathcal{Z}(L)_{\mathbb{F}}\coloneqq\mathcal{Z}(L)\times_{W}\mathbb{F}$ is
formally smooth over $\mathbb{F}$ of relative dimension $n-r-1$ at $z$ by
Lemma 4.1.1. Let
$\mathcal{N}^{u}_{\mathbb{F}}=\mathcal{N}^{u}\times_{W}\mathbb{F}$. Let
$l\in\overline{\mathbf{D}}$ be an element such that its image $\overline{l}$
in $\overline{\mathbf{D}}_{\mathbb{F}}$ is a generator of the line
$\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}$. Let
$\mathbb{F}[\epsilon]=\mathbb{F}[X]/X^{2}$.

By Schlessinger’s criterion in [Sch68, Theorem 2.11], the tangent space
$\textup{Tgt}_{z}(\mathcal{N}^{u}_{\mathbb{F}})$ of
$\mathcal{N}^{u}_{\mathbb{F}}$ at $z$ can be identified with the set
$\widehat{\mathcal{N}}^{u}_{z}(\mathbb{F}[\epsilon])$, this set has a natural
bijection to the set of lines in
$\overline{\mathbf{D}}\otimes_{W}\mathbb{F}[\epsilon]$ which lifts the line
$\mathbb{F}\cdot\overline{l}$ in $\overline{\mathbf{D}}_{\mathbb{F}}$ by Lemma
3.4.2, therefore any such line has a generator of the form
$\overline{l}+\epsilon\cdot w$ for some element
$w\in\overline{\mathbf{D}}_{\mathbb{F}}$. Two elements
$\overline{l}+\epsilon\cdot w$ and $\overline{l}+\epsilon\cdot w^{\prime}$
generate the same line if and only if
$w^{\prime}-w^{\prime}\in\mathbb{F}\cdot\overline{l}$, hence
$\textup{Tgt}_{z}(\mathcal{N}^{u}_{\mathbb{F}})\simeq\overline{\mathbf{D}}_{\mathbb{F}}/\mathbb{F}\cdot\overline{l}$,
and
$\textup{dim}_{\mathbb{F}}\textup{Tgt}_{z}(\mathcal{N}^{u}_{\mathbb{F}})=n-1$.

The tangent space $\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})$ of
$\mathcal{Z}(L)_{\mathbb{F}}$ at $z$ can be identified with the set
$\widehat{\mathcal{Z}(L)}_{z}(\mathbb{F}[\epsilon])$. For any $x\in L$, let
$\overline{x_{\textup{crys,z}}}$ be the image of $x_{\textup{crys,z}}$ in
$\mathbf{D}_{\mathbb{F}}$. The set
$\widehat{\mathcal{Z}(L)}_{z}(\mathbb{F}[\epsilon])$ has a natural bijection
to the set of lines in $\overline{\mathbf{D}}\otimes_{W}\mathbb{F}[\epsilon]$
which lifts the line $\mathbb{F}\cdot\overline{l}$ in
$\overline{\mathbf{D}}_{\mathbb{F}}$ and orthogonal to
$\overline{x_{\textup{crys},z}}$ for any $x\in L$. Let
$\overline{l}+\epsilon\cdot w$ be a generator of a line in the set
$\widehat{\mathcal{Z}(L)}_{z}(\mathbb{F}[\epsilon])$, then
$\langle\overline{x_{\textup{crys},z}},\overline{l}+\epsilon\cdot
w\rangle=\epsilon\langle\overline{x_{\textup{crys},z}},w\rangle=0$, hence
$\langle\overline{x_{\textup{crys},z}},w\rangle=0$, therefore
$\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})$ is the subspace of
$\overline{\mathbf{D}}_{\mathbb{F}}/\mathbb{F}\cdot\overline{l}$ orthogonal to
the image of $L_{W}$ in $\mathbf{D}_{\mathbb{F}}$ under the pairing
$\langle\cdot,\cdot\rangle_{\mathbb{F}}$. Let
$\overline{L}_{W}\coloneqq(i_{\textup{crys},z}(L_{W})+p\mathbf{D})/p\mathbf{D}$
be the image of $L_{W}$ in $\mathbf{D}_{\mathbb{F}}$. Since the pairing
$\langle\cdot,\cdot\rangle_{\mathbb{F}}$ is non-degenerate, we have

(1)
$\textup{dim}_{\mathbb{F}}\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})=n-1-\textup{dim}_{\mathbb{F}}(\overline{L}_{W}).$

therefore $\mathcal{Z}(L)$ is formally smooth over $W$ of relative dimension
$n-r-1$ at $z$ if and only if $\textup{dim}_{\mathbb{F}}(\overline{L}_{W})=r$,
which is equivalent to the fact that the map
$\overline{i_{\textup{crys},z}}:L_{\mathbb{F}}\rightarrow\mathbf{D}_{\mathbb{F}}$
is injective, i.e., the isometric map $i_{\textup{crys},z}$ is primitive. ∎

####  4.1.2. The GSpin case

For a point $z\in\mathcal{N}(\mathbb{F})$, let $\widehat{\mathcal{N}}^{o}_{z}$
be the completion of the formal scheme $\mathcal{N}^{o}$ at $z$. Let
$\mathbf{V}=\mathbf{V}_{\textup{crys},z}(W)$ be the $W$-module defined in
$\S$3.3. Let $L\subset\mathbb{V}^{o}$ be an $\mathcal{O}_{F}$-lattice and
suppose that $z\in\mathcal{Z}(L)(\mathbb{F})$, then the crystalline
realization $x_{\textup{crys},z}$ belongs to $\mathbf{V}$, therefore we have
an isometric map $i_{\textup{crys,z}}:L\rightarrow\mathbf{V}$. We say the
isometric map $i_{\textup{crys},z}$ is primitive at $z$ if the induced map
$\overline{i_{\textup{crys},z}}:L_{\mathbb{F}}\rightarrow\mathbf{V}_{\mathbb{F}}$
is injective.

######  Lemma 4.1.3.

Let $L\subset\mathbb{V}^{o}$ be an $\mathcal{O}_{F}$-lattice of rank $r\geq 1$
and $z\in\mathcal{Z}(L)(\mathbb{F})$, then $\mathcal{Z}(L)$ is formally smooth
over $W$ of relative dimension $n-r-1$ at $z$ if and only if the following two
assertions hold,
(i) The isometric map $i_{\textup{crys},z}$ is primitive at $z$;
(ii) There exists a lift of $z$ to
$z^{\prime}\in\mathcal{Z}(L)(W/\varpi^{2})$.

###### Proof.

The special cycle $\mathcal{Z}(L)$ is cut out by $r$ equations in the
deformation space
$\widehat{\mathcal{N}}^{o}_{z}\simeq\textup{Spf}\,\mathcal{O}_{\mathcal{N},z}$
by Proposition 2.3.3, therefore the special cycle $\mathcal{Z}(L)$ is formally
smooth over $W$ of relative dimension $n-r-1$ at $z$ if and only if
$\mathcal{Z}(L)_{\mathbb{F}}\coloneqq\mathcal{Z}(L)\times_{W}\mathbb{F}$ is
formally smooth over $\mathbb{F}$ of relative dimension $n-r-1$ at $z$ by
Lemma 4.1.1. Let
$\mathcal{N}^{o}_{\mathbb{F}}=\mathcal{N}^{o}\times_{W}\mathbb{F}$. Let
$l\in\mathbf{V}$ be an isotropic element such that its image
$\overline{l}\in\mathbf{V}_{\mathbb{F}}$ generates the line
$\textup{Fil}^{1}(\mathbf{V}_{\mathbb{F}})$. Let $(\cdot,\cdot)$ be the
bilinear form on $\mathbf{V}$ and $(\cdot,\cdot)_{\mathbb{F}}$ be the bilinear
form on $\mathbf{V}_{\mathbb{F}}$. Let
$\mathbb{F}[\epsilon]=\mathbb{F}[X]/X^{2}$.

By Schlessinger’s criterion in [Sch68, Theorem 2.11], the tangent space
$\textup{Tgt}_{z}(\mathcal{N}^{o}_{\mathbb{F}})$ (resp.
$\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})$) of
$\mathcal{N}^{o}_{\mathbb{F}}$ (resp. $\mathcal{Z}(L)_{\mathbb{F}}$) at $z$
can be identified with the set
$\widehat{\mathcal{N}}_{z}^{o}(\mathbb{F}[\epsilon])$, this set has a natural
bijection to the set of isotropic lines in
$\mathbf{V}\otimes_{W}\mathbb{F}[\epsilon]$ which lifts the line
$\mathbb{F}\cdot\overline{l}$ in $\mathbf{V}_{\mathbb{F}}$ by Lemma 3.4.3,
therefore any such line has a generator of the form
$\overline{l}+\epsilon\cdot w$ for some $w\in\mathbf{V}_{\mathbb{F}}$, note
that

$0=q_{\mathbf{V}\otimes_{W}\mathbb{F}[\epsilon]}(\overline{l}+\epsilon\cdot
w)=q_{\mathbf{V}_{\mathbb{F}}}(\overline{l})+\epsilon(\overline{l},w)_{\mathbb{F}}=\epsilon(\overline{l},w)_{\mathbb{F}},$

hence the fact that the line generated by $\overline{l}+\epsilon\cdot w$ is
isotropic is equivalent to the vector $w\in\mathbf{V}_{\mathbb{F}}$ is
orthogonal to $\overline{l}$. Two elements $\overline{l}+\epsilon\cdot w$ and
$\overline{l}+\epsilon\cdot w^{\prime}$ generate the same line if and only if
$w^{\prime}-w\in\mathbb{F}\cdot\overline{l}$, hence
$\textup{Tgt}_{z}(\mathcal{N}^{o}_{\mathbb{F}})\simeq\\{\overline{l}\\}^{\bot}/\mathbb{F}\cdot\overline{l}$,
and
$\textup{dim}_{\mathbb{F}}\textup{Tgt}_{z}(\mathcal{N}^{o}_{\mathbb{F}})=n-1$.
There is a bilinear pairing on the space
$\\{\overline{l}\\}^{\bot}/\mathbb{F}\cdot\overline{l}$ which is induced by
$(\cdot,\cdot)_{\mathbb{F}}$, we use $\overline{(\cdot,\cdot)_{\mathbb{F}}}$
to denote it. It is a non-degenerate since the bilinear pairing
$(\cdot,\cdot)_{\mathbb{F}}$ on $\mathbf{V}_{\mathbb{F}}$ is non-degenerate.

The tangent space $\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})$ of
$\mathcal{Z}(L)_{\mathbb{F}}$ at $z$ can be identified with the set
$\widehat{\mathcal{Z}(L)}_{z}(\mathbb{F}[\epsilon])$. For any $x\in L$, let
$\overline{x_{\textup{crys,z}}}$ be the image of $x_{\textup{crys,z}}$ in
$\mathbf{V}_{\mathbb{F}}$. The set
$\widehat{\mathcal{Z}(L)}_{z}(\mathbb{F}[\epsilon])$ has a natural bijection
to the set of isotropic lines in $\mathbf{V}\otimes_{W}\mathbb{F}[\epsilon]$
which lifts the line $\mathbb{F}\cdot\overline{l}$ in
$\mathbf{V}_{\mathbb{F}}$ and orthogonal to $\overline{x_{\textup{crys},z}}$
for any $x\in L$. Let $\overline{l}+\epsilon\cdot w$ be a generator of a line
in the set $\widehat{\mathcal{Z}(L)}_{z}(\mathbb{F}[\epsilon])$, then
$(\overline{x_{\textup{crys},z}},\overline{l}+\epsilon\cdot
w)=\epsilon(\overline{x_{\textup{crys},z}},w)=0$, hence
$(\overline{x_{\textup{crys},z}},w)=0$, therefore
$\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})$ is the subspace of
$\\{\overline{l}\\}^{\bot}/\mathbb{F}\cdot\overline{l}$ orthogonal to the
image of $L_{W}$ in $\mathbf{V}_{\mathbb{F}}$ under the pairing
$(\cdot,\cdot)_{\mathbb{F}}$, let
$\overline{L}_{W}\coloneqq(i_{\textup{crys},z}(L_{W})+p\mathbf{V})/p\mathbf{V}$
be the image of $L_{W}$ in $\mathbf{V}_{\mathbb{F}}$, it is contained in the
subspace $\\{\overline{l}\\}^{\bot}$ of $\mathbf{V}_{\mathbb{F}}$, we have
$\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})=\left((\overline{L}_{W}+\mathbb{F}\cdot\overline{l})/\mathbb{F}\cdot\overline{l}\right)^{\bot}\subset\\{\overline{l}\\}^{\bot}/\mathbb{F}\cdot\overline{l}$.
Since the pairing $\overline{(\cdot,\cdot)_{\mathbb{F}}}$ is non-degenerate on
$\\{\overline{l}\\}^{\bot}/\mathbb{F}\cdot\overline{l}$,
$\textup{dim}_{\mathbb{F}}\,\textup{Tgt}_{z}(\mathcal{Z}(L)_{\mathbb{F}})=n-1-\textup{dim}_{\mathbb{F}}\,\left((\overline{L}_{W}+\mathbb{F}\cdot\overline{l})/\mathbb{F}\cdot\overline{l}\right)$,
therefore $\mathcal{Z}(L)$ is formally smooth over $W$ of relative dimension
$n-r-1$ at $z$ if and only if
$\textup{dim}_{\mathbb{F}}\,\left((\overline{L}_{W}+\mathbb{F}\cdot\overline{l})/\mathbb{F}\cdot\overline{l}\right)=r$,
which is further equivalent to $\textup{dim}_{\mathbb{F}}\,\overline{L}_{W}=r$
and $\overline{l}\notin\overline{L}_{W}$. The condition
$\textup{dim}_{\mathbb{F}}\,\overline{L}_{W}=r$ is equivalent to (i) which
says that the isometric map $i_{\textup{crys,z}}$ is primitive at $z$.

Let’s assume (i) and (ii) are true, we want to show that the element
$\overline{l}$ doesn’t belong to the space $\overline{L}_{W}$. Let
$z^{\prime}\in\mathcal{Z}(L)(W/\varpi^{2})$ be a lift of $z$ to
$W/\varpi^{2}$, and $\tilde{z}\in\widehat{\mathcal{N}}_{z}(W)$ be a lift of
$z^{\prime}$ to $W$. The point $\tilde{z}$ corresponds to an isotropic line
$L^{1}$, let $l\in\mathbf{V}$ be a generator of $L^{1}$ whose image in
$\mathbf{V}_{\mathbb{F}}$ is $\overline{l}$. Let’s assume the contrary, then
there exists $x\in L_{W}$ and $v\in\mathbf{V}$ such that

(2) $l=x_{\textup{crys},z}+\varpi\cdot v\in\mathbf{V}.$

Let $\\{x_{i}\\}_{i=1}^{r}$ be an $\mathcal{O}_{F}$-basis of $L$ and
$x=\sum\limits_{i=1}^{r}a_{i}x_{i}$ for some $a_{i}\in W$, then
$\overline{l}=\sum\limits_{i=1}^{r}\overline{a_{i}}\cdot\overline{x_{i,\textup{crys},z}}\neq
0$, hence there exists at least one $i$ such that $a_{i}\in W^{\times}$ and
$x_{i,\textup{crys},z}\notin\varpi\mathbf{V}$. Let $\Phi$ be the Frobenius
action on $\mathbf{V}$. If $v\in L^{0}+\varpi\mathbf{V}$, then
$\Phi(v)\in\mathbf{V}$ by Lemma 3.3.1, the same lemma also implies that
$\Phi(l)\in\varpi\mathbf{V}$, therefore
$\Phi(x_{\textup{crys},z})=\Phi(l)-\varpi\Phi(v)\in\varpi\mathbf{V}$. However,

$\Phi(x_{\textup{crys},z})=\sum\limits_{i=1}^{r}\sigma(a_{i})\cdot\Phi(x_{i,\textup{crys},z})=\sum\limits_{i=1}^{r}\sigma(a_{i})\cdot
x_{i,\textup{crys},z},$

hence $\Phi(i_{\textup{crys},z}(x))\notin\varpi\mathbf{V}$ because there
exists an integer $1\leq i\leq r$ such that $\sigma(a_{i})\in W^{\times}$ and
$x_{i,\textup{crys},z}\notin\varpi\mathbf{V}$, hence $v\notin
L^{0}+\varpi\mathbf{V}$, therefore $(l,v)\in W^{\times}$ by Lemma 3.3.2.

Recall that $\nu_{\varpi}$ is the $\varpi$-adic valuation on $K$. The point
$z^{\prime}\in\mathcal{Z}(L)(W/\varpi^{2})$ is a lift of $z$ to $W/\varpi^{2}$
implies that $\nu_{\varpi}((l,x_{i,\textup{crys},z}))\geq 2$ by Lemma 3.4.3,
hence
$\nu_{\varpi}((l,x_{\textup{crys},z}))=\nu_{\varpi}(\sum\limits_{i=1}^{r}a_{i}(l,x_{i,\textup{crys},z}))\geq
2$. However, the pairing $(l,x_{\textup{crys},z})=(l,l-\varpi\cdot
v)=-\varpi(l,v)$ by (2), hence $\nu_{\varpi}((l,x_{\textup{crys},z}))=1$ since
$(l,v)\in W^{\times}$ by previous discussion, this is a contradiction.
Therefore the element $\overline{l}$ doesn’t belong to the space
$\overline{L}_{W}$ if (i) and (ii) are true. ∎

###  4.2. Regularity of the difference divisor

######  Lemma 4.2.1.

Let $n\geq 2$ be an integer, let
$R=\mathcal{O}[[t_{1},\cdot\cdot\cdot,t_{n-1}]]$ where $\mathcal{O}$ is a
discrete valuation ring of characteristic $(0,p)$ with uniformizer $\pi$ and
$\pi$-adic valuation $\nu_{\pi}$, the ring $R$ has maximal ideal
$\mathfrak{m}_{R}\coloneqq(\pi,t_{1},\cdot\cdot\cdot,t_{n-1})$. In the
following the symbol $(\textup{unit})$ means an element in $R^{\times}$.
$\bullet$ Let $g\in\mathfrak{m}_{R}$. If for any continuous ring homomorphism
$f:R\rightarrow\mathcal{O}$, we have $\nu_{\pi}(f(g))=1$, then
$g\equiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$.
$\bullet$ Let $d\in\mathfrak{m}_{R}\backslash\mathfrak{m}_{R}^{2}$, let $h$ be
an element in $\mathfrak{m}_{R}$ such that
$h\nequiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$.
Then there exists a continuous ring homomorphism $f:R\rightarrow\mathcal{O}$
such that $f(d)\neq 0$ and $\nu_{\pi}(f(h))\geq 2$.

###### Proof.

For the first assertion, let $g\equiv
a_{0}\pi+\sum\limits_{j=1}^{n-1}a_{j}t_{j}\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$
where $a_{j}\in\mathcal{O}$ for every $0\leq j\leq n-1$. If
$\nu_{\pi}(a_{0})\geq 1$, consider the continuous homomorphism
$f:R\rightarrow\mathcal{O}$ such that $f(t_{j})=\pi^{2}$ for every $0\leq
j\leq n-1$, then $\nu_{\pi}(f(g))\geq 2$, which is a contradiction, therefore
$\nu_{\pi}(a_{0})=0$. If
$g\nequiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$,
then there exists at least one $j\geq 1$ such that $\nu_{\pi}(a_{j})=0$.
Suppose that $\nu_{\pi}(a_{j_{0}})=0$ for some $j_{0}\geq 1$, consider the
continuous homomorphism $f:R\rightarrow\mathcal{O}$ such that
$f(t_{j_{0}})=-a_{j_{0}}^{-1}a_{0}\pi$ and $f(t_{j})=\pi^{2}$ for $j\neq
j_{0}$, then $\nu_{\pi}(f(g))\geq 2$, which is a contradiction, hence
$g\equiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$.

For the second assertion, we consider the following two cases.

Case 1:
$d\equiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$.
Since
$h\nequiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$,
there exists a continuous ring homomorphism $f:R\rightarrow\mathcal{O}$ such
that $\nu_{\pi}(f(h))\geq 2$ by the proof of the first assertion, for this
$f$, we have $\nu_{\pi}(f(d))=1$, hence $f(d)\neq 0$.

Case 2:
$d\nequiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$,
then we can choose another system of uniformizers $t_{1},\cdots,t_{n-1}$ such
that $d=t_{1}$. Let $h\equiv
b_{0}\pi+\sum\limits_{j=1}^{n-1}b_{j}t_{j}\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$
where $b_{j}\in\mathcal{O}$ for every $0\leq j\leq n-1$. If
$\nu_{\pi}(b_{0})\geq 1$, consider the continuous homomorphism
$f:R\rightarrow\mathcal{O}$ such that $f(t_{j})=\pi^{2}$ for every $0\leq
j\leq n-1$, then $\nu_{\pi}(f(h))\geq 2$ and $f(d)=f(t_{1})\neq 0$. If
$\nu_{\pi}(b_{0})=0$, then there exists at least one $j\geq 1$ such that
$\nu_{\pi}(b_{j})=0$ since
$h\nequiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$.
Suppose that $\nu_{\pi}(b_{j_{0}})=0$ for some $j_{0}\geq 1$, consider the
continuous homomorphism $f:R\rightarrow\mathcal{O}$ such that
$f(t_{j_{0}})=-b_{j_{0}}^{-1}b_{0}\pi$ and $f(t_{j})=\pi^{2}$ for $j\neq
j_{0}$, then $\nu_{\pi}(f(g))\geq 2$ and $f(d)=f(t_{1})\neq 0$. ∎

######  Lemma 4.2.2.

Let $n\geq 2$ be an integer, let
$R=\mathcal{O}[[t_{1},\cdot\cdot\cdot,t_{n-1}]]$ where $\mathcal{O}$ is a
discrete valuation ring of characteristic $(0,p)$ with uniformizer $\pi$ and
$\pi$-adic valuation $\nu_{\pi}$. Let $(d_{a})_{a\geq 0}$ be a sequence of
elements in $R$ such that
$d_{a}\in\mathfrak{m}_{R}\coloneqq(\pi,t_{1},\cdot\cdot\cdot,t_{n-1})$ for any
$a\geq 0$ and $d_{0}\in\mathfrak{m}_{R}\backslash\mathfrak{m}_{R}^{2}$. Let
$f_{a}=\prod\limits_{i=0}^{a}d_{i}$ and
$\mathcal{Z}(f_{a})\coloneqq\textup{Spf}\,R/f_{a}$ be the closed formal
subscheme of $\textup{Spf}\,R$. For any morphism
$z:\textup{Spf}\,\mathcal{O}\rightarrow\textup{Spf}\,R$, we use $z^{\sharp}$
to denote the corresponding ring homomorphism $R\rightarrow\mathcal{O}$. If
for any $a\geq 1$ and any morphism
$z:\textup{Spf}\,\mathcal{O}\rightarrow\textup{Spf}\,R$, there exists a
Cartesian diagram

$\textstyle{\textup{Spec}\,\mathcal{O}/(\pi^{a}\cdot
z^{\sharp}(d_{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{Spf}\,\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z}$$\textstyle{\mathcal{Z}(f_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{Spf}\,R.}$

then
$d_{a}\equiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$
for all $a\geq 1$.

###### Proof.

Let’s assume the contrary that there exists $a\geq 1$ such that
$d_{a}\nequiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$,
let $k\geq 1$ be the least integer such that
$d_{i}\equiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$
for all $1\leq i<k$ and
$d_{k}\nequiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$,
then by Lemma 4.2.1 there exists a morphism
$z:\textup{Spf}\,\mathcal{O}\rightarrow\textup{Spf}\,R$ such that
$z^{\sharp}(d_{0})\neq 0$ and $\nu_{\pi}(z^{\sharp}(d_{k}))\geq 2$, then
$\mathcal{Z}(f_{k})\times_{\textup{Spf}\,R,z}\textup{Spf}\,\mathcal{O}$ is cut
out in $\textup{Spf}\,\mathcal{O}$ by $(\pi^{m})$ for some integer $m$ such
that $k+\nu_{\pi}(z^{\sharp}(d_{0}))+1\leq m<\infty$, hence the fiber product
$\mathcal{Z}(f_{k})\times_{\textup{Spf}\,R,z}\textup{Spf}\,\mathcal{O}$ is not
isomorphic to $\textup{Spec}\,\mathcal{O}/(\pi^{k}\cdot z^{\sharp}(d_{0}))$,
which is a contradiction, therefore such $k$ doesn’t exist, hence
$d_{a}\equiv(\textup{unit})\cdot\pi\,\,(\textup{mod}\,\,\mathfrak{m}_{R}^{2})$
for all $a\geq 1$. ∎

######  Lemma 4.2.3.

Let $x\in\mathbb{V}^{u}$ (resp. $x\in\mathbb{V}^{o}$) be a special quasi-
homomorphism such that $(x,x)\in\mathcal{O}_{E}\backslash\\{0\\}$ (resp.
$q_{\mathbb{V}^{o}}(x)\in\mathcal{O}_{F}\backslash\\{0\\}$). Let
$z\in\mathcal{Z}(x)(\mathbb{F})$ be a point, let
$\tilde{z}\in\mathcal{N}^{u}(W)$ (resp. $\mathcal{N}^{o}(W)$) be a lift of $z$
to $W$ and $\tilde{z}^{\sharp}:\mathcal{O}_{\mathcal{N}^{u},z}$ (resp.
$\mathcal{O}_{\mathcal{N}^{o},z}$) $\rightarrow W$ be the corresponding
continuous homomorphism. Let $f_{x,z}\in\mathcal{O}_{\mathcal{N}^{u},z}$
(resp. $\mathcal{O}_{\mathcal{N}^{o},z}$) be the equation of the Cartier
divisor $\mathcal{Z}(x)$ at $z$.
$\bullet$ In the unitary case, let $\mathbf{D}$ and $\overline{\mathbf{D}}$ be
the two $W$-lattices of rank $n$ corresponding to $z$ defined in $\S$3.2, let
$\langle\cdot,\cdot\rangle:\mathbf{D}\times\overline{\mathbf{D}}\rightarrow W$
be the pairing induced by the $p$-principal polarization. The lift $\tilde{z}$
corresponds to a line $L^{1}\subset\overline{\mathbf{D}}$ which has a
generator $l$, then

(3) $(\tilde{z}^{\sharp}(f_{x,z}))=(\langle
x_{\textup{crys,z}},l\rangle)\subset W.$

$\bullet$ In the GSpin case, let $\mathbf{V}=\mathbf{V}_{\textup{crys},z}(W)$
be the $W$-module defined in $\S$3.3, let
$(\cdot,\cdot):\mathbf{V}\times\mathbf{V}\rightarrow W$ be the bilinear
pairing induced by the quadratic form $q_{\mathbf{V}}$ on $\mathbf{V}$. The
lift $\tilde{z}$ corresponds to an isotropic line $L^{1}\subset\mathbf{V}$
which has a generator $l$, then

(4) $(\tilde{z}^{\sharp}(f_{x,z}))=\left((x_{\textup{crys,z}},l)\right)\subset
W.$

###### Proof.

We only give the proof in the unitary case, the proof of the GSpin case is
similar. Let $\widehat{\mathcal{Z}(x)}_{z}$ be the completion of the special
cycle $\mathcal{Z}(x)$ at $z$, there exists a Cartesian diagram,

$\textstyle{\textup{Spec}\,W/(\tilde{z}^{\sharp}(f_{x,z}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{Spf}\,W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{z}}$$\textstyle{\widehat{\mathcal{Z}(x)}_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{N}}^{u}_{z}.}$

Let
$m=\nu_{\varpi}(\tilde{z}^{\sharp}(f_{x,z}))\in\mathbb{Z}\cup\\{\infty\\}$,
$t=\nu_{\varpi}(\langle
x_{\textup{crys},z},l\rangle)\in\mathbb{Z}\cup\\{\infty\\}$, the equality in
(3) is equivalent to $t=m$.

The morphism $W/(\tilde{z}^{\sharp}(f_{x,z}))\rightarrow\mathbb{F}$ has a
natural divided power structure, therefore the fact that the special quasi-
homomorphism $x$ can be lifted to
$\textup{Spec}\,W/(\tilde{z}^{\sharp}(f_{x,z}))$ implies that $\langle
x_{\textup{crys},z},l\rangle\equiv 0\,\,(\textup{mod}\,\,\varpi^{m})$ by Lemma
3.4.2, hence $t\geq m$.

On the other hand, the morphism $W/(\varpi^{t})\rightarrow\mathbb{F}$ has a
natural divided power structure, the quasi-homomorphism $x$ can be lifted to
$W/(\varpi^{t})$ because $\langle x_{\textup{crys},z},l\rangle\equiv
0\,\,(\textup{mod}\,\,\varpi^{t})$ by Lemma 3.4.2, therefore we have a
morphism
$\textup{Spec}\,W/(\varpi^{t})\rightarrow\widehat{\mathcal{Z}(x)}_{z}$ which
also factors through $\tilde{z}$, hence there is a morphism
$\textup{Spec}\,W/(\varpi^{t})\rightarrow\textup{Spec}\,W/(\tilde{z}^{\sharp}(f_{x,z}))=\textup{Spec}\,W/(\varpi^{m})$,
so $m\geq t$. ∎

######  Theorem 4.2.4.

Let $x\in\mathbb{V}^{u}$ (resp. $x\in\mathbb{V}^{o}$) be a special quasi-
homorphism such that $(x,x)\in\mathcal{O}_{E}\backslash\\{0\\}$ (resp.
$q_{\mathbb{V}^{o}}(x)\in\mathcal{O}_{F}\backslash\\{0\\}$). Let
$z\in\mathcal{N}^{u}(\mathbb{F})$ (resp. $z\in\mathcal{N}^{o}(\mathbb{F})$) be
a point such that $z\in\mathcal{Z}(x)(\mathbb{F})$ but
$x\notin\mathcal{Z}(\varpi^{-1}x)(\mathbb{F})$.
$\bullet$ If there is no lift of $z$ to
$z^{\prime}\in\mathcal{Z}(x)(W/\varpi^{2})$, then
$z\in\mathcal{D}(\varpi^{a}x)(\mathbb{F})$ for any integer $a\geq 0$, and the
difference divisor $\mathcal{D}(\varpi^{a}x)$ is regular but not formally
smooth over $W$ at $z$.
$\bullet$ If there exists a lift of $z$ to
$z^{\prime}\in\mathcal{Z}(x)(W/\varpi^{2})$, then the special divisor
$\mathcal{Z}(x)$ is formally smooth over $W$ at $z$, and
$z\in\mathcal{D}(\varpi^{a}x)(\mathbb{F})$ for any integer $a\geq 0$, and the
difference divisor $\mathcal{D}(\varpi^{a}x)$ is regular at $z$. Furthermore,
the difference divisor $\mathcal{D}(x)$ is formally smooth over $W$ at $z$,
while $\mathcal{D}(\varpi^{a}x)$ is not formally smooth over $W$ at $z$ for
$a\geq 1$.

Proof of the unitary case. Let $\mathbf{D}\coloneqq\mathbb{D}_{0}(X)$ and
$\overline{\mathbf{D}}\coloneqq\mathbb{D}_{1}(X)$ be the two $W$-lattices of
rank $n$ corresponding to $z$ defined in $\S$3.2, let
$\langle\cdot,\cdot\rangle:\mathbf{D}\times\overline{\mathbf{D}}\rightarrow W$
be the pairing induced by the $p$-principal polarization. Let
$\tilde{z}\in\mathcal{N}^{u}(W)$ be an arbitrary lift of $z$ to $W$, it
corresponds to a line $L^{1}\subset\overline{\mathbf{D}}$ which lifts
$\textbf{F}^{1}D_{\mathbb{F}}\cap\overline{\mathbf{D}}_{\mathbb{F}}$, it also
corresponds to a continuous homomorphism
$\tilde{z}^{\sharp}:\mathcal{O}_{\mathcal{N}^{u},z}\simeq
W[[t_{1},\cdots,t_{n-1}]]\rightarrow W$. Let $l$ be a generator of the line
$L^{1}$ and $f_{\varpi^{a}x}$ be the equation defined by the quasi-isogeny
$\varpi^{a}x$ in the ring $\mathcal{O}_{\mathcal{N}^{u},z}$ for any $a\geq 0$,
we have the following equality of ideals of $W$ by the formula (3) in Lemma 4,

$(\tilde{z}^{\sharp}(f_{\varpi^{a}x}))=(\langle\varpi^{a}\cdot
x_{\textup{crys,z}},l\rangle)=(\varpi^{a}\cdot\langle
x_{\textup{crys,z}},l\rangle)=(\varpi^{a}\cdot\tilde{z}^{\sharp}(f_{x})).$

Let $\mathfrak{m}_{z}=(\varpi,t_{1},\cdots,t_{n-1})$ be the maximal ideal of
$\mathcal{O}_{\mathcal{N}^{u},z}$. We first prove that
$d_{x}=f_{x}\in\mathfrak{m}_{z}\backslash\mathfrak{m}_{z}^{2}$.
$\bullet$ If there is no lift of $z$ to
$z^{\prime}\in\mathcal{Z}(x)(W/\varpi^{2})$, then $\nu_{\varpi}(\langle
x_{\textup{crys,z}},l\rangle)=1$, because otherwise $\nu_{\varpi}(\langle
x_{\textup{crys,z}},l\rangle)\geq 2$, so $z$ can be lifted to some
$z^{\prime}\in\mathcal{Z}(x)(W/\varpi^{2})$, therefore
$\nu_{\varpi}(\tilde{z}^{\sharp}(f_{x}))=1$ for any continuous homomorphism
$\tilde{z}^{\sharp}:W[[t_{1},\cdots,t_{n-1}]]\rightarrow W$, hence
$f_{x}\equiv(\textup{unit})\cdot\varpi\,\,(\textup{mod}\,\,\mathfrak{m}_{z}^{2})$
by Lemma 4.2.1. Note that $z\notin\mathcal{Z}(\varpi^{-1}x)$ implies that
$f_{\varpi^{-1}x}=1$ by our convention, then
$d_{x}=f_{x}\equiv(\textup{unit})\cdot\varpi\,\,(\textup{mod}\,\,\mathfrak{m}_{z}^{2})$.
$\bullet$ If there exists a lift of $z$ to
$z^{\prime}\in\mathcal{Z}(x)(W/\varpi^{2})$, let
$\tilde{z}\in\mathcal{N}^{u}(W)$ be a lift of $z^{\prime}$ to $W$ whose
corresponding line in $\overline{\mathbf{D}}$ has a generator $l$, then
$\nu_{\varpi}(\langle x_{\textup{crys},z},l\rangle)\geq 2$. We must have
$x_{\textup{crys},z}\notin\varpi\mathbf{D}$ because otherwise
$\varpi^{-1}x_{\textup{crys},z}\in\mathbf{D}$ and
$\nu_{\varpi}(\langle\varpi^{-1}x_{\textup{crys},z},l\rangle)\geq 1$, hence
$z\in\mathcal{Z}(\varpi^{-1}x)(\mathbb{F})$ by Lemma 3.4.2, which is a
contradiction, hence $x_{\textup{crys},z}\notin\varpi\mathbf{D}$. The element
$x_{\textup{crys},z}$ doesn’t belong to the lattice $\varpi\mathbf{D}$ is
equivalent to the fact that the isometric map
$i_{\textup{crys,z}}:\mathcal{O}_{E}\cdot x\rightarrow\mathbf{D}$ is primitive
at $z$, therefore by Lemma 4.1.2 the special cycle $\mathcal{Z}(x)$ is
formally smooth over $W$ of relative dimension $n-2$ at $z$, so
$d_{x}=f_{x}\in\mathfrak{m}_{z}\backslash\mathfrak{m}_{z}^{2}$.

Let $\widehat{\mathcal{N}}^{u}_{z}$ (resp.
$\widehat{\mathcal{Z}(\varpi^{a}x)}_{z}$) be the completion of the formal
scheme $\mathcal{N}^{u}$ (resp. $\mathcal{Z}(\varpi^{a}x)$) at $z$, there
exists a Cartesian diagram by (3),

$\textstyle{\textup{Spec}\,W/(\varpi^{a}\cdot\tilde{z}^{\sharp}(f_{x}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{Spf}\,W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{z}}$$\textstyle{\widehat{\mathcal{Z}(\varpi^{a}x)}_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{N}}^{u}_{z}.}$

Recall that by the definition of difference divisors, we have
$f_{\varpi^{a}x}=\prod\limits_{i=0}^{a}d_{\varpi^{i}x}$ for any integer $a\geq
0$, hence Lemma 4.2.2 implies that
$d_{\varpi^{a}x}\equiv(\textup{unit})\cdot\varpi\,\,(\textup{mod}\,\,\mathfrak{m}_{z}^{2})$
for any $a\geq 1$ since
$d_{x}\in\mathfrak{m}_{z}\backslash\mathfrak{m}_{z}^{2}$. ∎

Proof of the GSpin case. The proof is almost identical to the unitary case, we
use the same notation as $\S$3.3 and replace the lattice $\mathbf{D}$ by the
lattice $\mathbf{V}$, replace the pairing
$\langle\cdot,\cdot\rangle:\mathbf{D}\times\overline{\mathbf{D}}\rightarrow W$
by the symmetric pairing $(\cdot,\cdot):\mathbf{V}\times\mathbf{V}\rightarrow
W$ induced by the quadratic form $q_{\mathbf{V}}$, the element
$l\in\mathbf{V}$ is taken to be the generator of the isotropic line
$L^{1}\subset\mathbf{V}$ which corresponds to a lift of $z$ to $W$ as in
$\S$3.3.1.

######  Corollary 4.2.5.

Let $x\in\mathbb{V}^{u}$ (resp. $x\in\mathbb{V}^{o}$) be an element such that
$(x,x)\in\mathcal{O}_{E}\backslash\\{0\\}$ (resp.
$q_{\mathbb{V}^{o}}(x)\in\mathcal{O}_{F}\backslash\\{0\\}$), then
$\mathcal{D}(x)(\mathbb{F})=\mathcal{Z}(x)(\mathbb{F})$, and the difference
divisor $\mathcal{D}(x)$ is regular. Moreover, the difference divisor
$\mathcal{D}(x)$ is formally smooth over $W$ at a point
$z\in\mathcal{Z}(x)(\mathbb{F})$ if and only if
$z\notin\mathcal{Z}(\varpi^{-1}x)(\mathbb{F})$ and there exists a lift of $z$
to $z^{\prime}\in\mathcal{Z}(x)(W/\varpi^{2})$.

###### Proof.

Obviously $\mathcal{D}(x)(\mathbb{F})\subset\mathcal{Z}(x)(\mathbb{F})$.
Conversely, if $z\in\mathcal{Z}(x)(\mathbb{F})$, there exists an integer
$a\geq 0$ such that $z\in\mathcal{Z}(\varpi^{-a}x)(\mathbb{F})$ but
$z\notin\mathcal{Z}(\varpi^{-a-1}x)(\mathbb{F})$, Theorem 4.2.4 implies that
the local equation of $\mathcal{D}(x)$ at $z$ is nontrivial and
$\mathcal{D}(x)$ is regular at $z$, and $\mathcal{D}(x)$ is formally smooth
over $W$ at a point $z\in\mathcal{Z}(x)(\mathbb{F})$ if and only if
$z\notin\mathcal{Z}(\varpi^{-1}x)(\mathbb{F})$ and there exists a lift of $z$
to $z^{\prime}\in\mathcal{Z}(x)(W/\varpi^{2})$. ∎

## References

  * [BO78] Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978.
  * [HP17] Benjamin Howard and Georgios Pappas. Rapoport-Zink spaces for spinor groups. Compos. Math., 153(5):1050–1118, 2017.
  * [Kim18] Wansu Kim. Rapoport-Zink spaces of Hodge type. Forum Math. Sigma, 6:e8, 110, 2018.
  * [KR11] Stephen S. Kudla and Michael Rapoport. Special cycles on unitary Shimura varieties I. Unramified local theory. Invent. Math., 184(3):629–682, 2011.
  * [KR14] Stephen S. Kudla and Michael Rapoport. Special cycles on unitary Shimura varieties II. Global theory. J.Reine Angew. Math., 697:91–157, 2014.
  * [LZ17] Chao Li and Yihang Zhu. Remarks on the arithmetic fundamental lemma. Algebra Number Theory, 11(10):2425–2445, 2017.
  * [LZ22a] Chao Li and Wei Zhang. Kudla–Rapoport cycles and derivatives of local densities. J. Amer. Math. Soc., 35(3):705-797, 2022.
  * [LZ22b] Chao Li and Wei Zhang. On the arithmetic Siegel–Weil formula for GSpin Shimura varieties. Invent. Math., 228(3):1353-1460, 2022.
  * [Mih22] Andreas Mihatsch. Relative unitary RZ-spaces and the arithmetic fundamental lemma. J. Inst. Math. Jussieu., 21(1), 241-301, 2022.
  * [MP16] Keerthi Madapusi Pera. Integral canonical models for spin Shimura varieties. Compos. Math., 152(4):769–824, 2016.
  * [RTZ13] Michael Rapoport, Ulrich Terstiege and Wei Zhang, On the arithmetic fundamental lemma in the minuscule case, Compos. Math. 149(10), 1631–1666 (2013).
  * [RZ96] Michael Rapoport and Thomas Zink. Period Spaces for $p$-Divisible Groups. (AM-141). Princeton University Press, 1996.
  * [Sch68] Michael Schlessinger. Functors of Artin Rings. Trans. Amer. Math. Soc., 130(2): 208–22, 1968.
  * [Ter11] Ulrich Terstiege. Intersections of arithmetic Hirzebruch-Zagier cycles. Math. Ann., 349(1):161–213, 2011.
  * [Ter13a] Ulrich Terstiege, Intersections of special cycles on the Shimura variety for $GU(1,2)$, J.Reine Angew. Math., 684, 113–164, 2013.
  * [Ter13b] Ulrich Terstiege. On the regularity of special difference divisors. C. R. Math. Acad. Sci. Paris, 351(3-4):107–109, 2013.
  * [Vol10] Inken Vollaard. The supersingular locus of the Shimura variety for GU(1, s). Canad. J. Math., 62(3):668–720, 2010.
  * [Zin01] Thomas Zink. Windows for displays of $p$-divisible groups. In: Moduli of Abelian Varieties (Text 1999), Prog. in Math., (195):491–518. Birkhäuser, Basel (2001).
  * [Zin02] Thomas Zink. The display of a formal $p$-divisible group. In: Cohomologies $p$-Adiques et Application Arithmétiques I. Astérisque, (278):127–248. Soc. Math. France, Paris (2002)