2.1 Quadratic forms and lattices

Let us fix an odd prime $p$ and consider a finite field extension $F/{\mathbb {Q}}_p$. Let ${\mathcal {O}}={\mathcal {O}}_F$ be the ring of integers and let $\pi$ be a uniformizer of ${\mathcal {O}}$. We will denote by $k={\mathcal {O}}/(\pi )$ the residue field and by $\bar k$ an algebraic closure of $k$. Denote also by $\breve F$ the completion of the maximal unramified extension of $F$ in an algebraic closure $\bar F$ and by $\breve O$ the integers of $\breve F$. We can assume $\bar k=\breve {\mathcal {O}}/(\pi )$.

Let $V$ be an $F$-vector space of dimension $d$ and fix

\[ \langle\,{,}\,\rangle: V \times V\to F, \]

a non-degenerate symmetric $F$-bilinear form. Set

\[ Q(v)=\tfrac{1}{2}\langle v, v\rangle, \]

for the associated quadratic form $Q: V\to F$. We assume that $d\geq 5$ throughout.

If ${\Lambda }\subset V$ is an ${\mathcal {O}}$-lattice, we set

\[ \Lambda^\vee=\{v\in V\mid \langle v, a\rangle\in {\mathcal{O}},\ \forall a\in \Lambda\} \]

for the dual of $\Lambda$. In what follows, we will fix a ‘vertex lattice’ $\Lambda$ in $V$. By definition this is an ${\mathcal {O}}$-lattice ${\Lambda }$ such that

\[ \pi\Lambda^\vee\subset \Lambda\subset \Lambda^\vee. \]

Set $\delta ({\Lambda }):={\rm length}_{\mathcal {O}}(\Lambda ^\vee /\Lambda )=\dim _k(\Lambda ^\vee /\Lambda )$, $\delta _*={\rm min}(\delta, d-\delta )$. We will often assume $\delta (\Lambda )\leq d/2$. Indeed, we can always replace the form by a multiple and exchange the roles of ${\Lambda }$ and ${\Lambda }^\vee$ to come to this situation. Then $\delta _*=\delta$.

The form $\langle \,{,}\,\rangle$ induces symmetric ${\mathcal {O}}$-bilinear forms

\[ \langle\,{,}\,\rangle: \Lambda\times\Lambda \to {\mathcal{O}},\quad \pi\langle\,{,}\,\rangle : \Lambda^\vee\times \Lambda^\vee \to {\mathcal{O}}. \]

We also obtain perfect symmetric $k$-bilinear forms

\[ \langle\,{,}\,\rangle_1: \Lambda/\pi\Lambda^\vee \times \Lambda/\pi\Lambda^\vee \to k,\quad \langle\,{,}\,\rangle_2: \Lambda^\vee/\Lambda \times \Lambda^\vee/\Lambda \to k, \]

given by $\langle x, y\rangle _1=\langle x, y \rangle \, {\rm mod}\, (\pi )$, $\langle x, y\rangle _2=\pi \langle x, y \rangle \, {\rm mod}\, (\pi )$.

2.2 Normal forms

As in [Reference Rapoport and ZinkRZ96, Appendix, Lemma A.25], we can write

\[ {\Lambda}=M\oplus N, \quad {\Lambda}^\vee=M\oplus \pi^{-1}N, \]

where $M$, $N$ are free ${\mathcal {O}}_{F}$-submodules such that $\langle M, N\rangle = 0$ and with the property that the form $\langle \,{,}\,\rangle$ is perfect on $M$ and $\pi ^{-1}\langle \,{,}\,\rangle$ is perfect on $N$. In fact, by [Reference Rapoport and ZinkRZ96, Appendix, Proposition A.21], after a finite unramified base change $F'/F$ we can find an ${\mathcal {O}}_{F'}$-basis $\{e_i\}$ of ${\Lambda }\otimes _{{\mathcal {O}}_F}{\mathcal {O}}_{F'}$, that is, write

\[ {\Lambda}_{{\mathcal{O}}_{F'}}= \Lambda\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_{F'} =\bigoplus_{i=1}^d {\mathcal{O}}_{F'}\cdot e_i \]

such that exactly one of the following cases occurs.

(1) $d=2n$, $\delta (\Lambda )=2r$:

\[ {\Lambda}_{{\mathcal{O}}_{F'}}=(e_1, \ldots, e_{n-r})\oplus (e_{n-r+1},\ldots , e_{n+r})\oplus (e_{n+r+1}, \ldots, e_d), \]

\[ M=(e_1, \ldots, e_{n-r})\oplus (e_{n+r+1}, \ldots, e_d),\quad N=(e_{n-r+1},\ldots , e_{n+r}), \]

\begin{align*} {\rm with}\quad \langle e_i , e_{d+1-j} \rangle&= \delta_{ij}, \quad \text{for } i< n-r+1 \text{ or } n+r< i, \\ \langle e_i , e_{d+1-j} \rangle&= \pi \delta_{ij}, \quad \text{for } n-r+1\leq i\leq n+r. \end{align*}

(2) $d=2n$, $\delta (\Lambda )=2r+1$:

\[ {\Lambda}_{{\mathcal{O}}_{F'}}=(e_1, \ldots, e_{n-r-1})\oplus (e_{n-r},\ldots , e_{n-1})\oplus (e_n, e_{n+1})\oplus (e_{n+2},\ldots , e_{n+r+1})\oplus (e_{n+r+2}, \ldots, e_d), \]

\[ M=(e_1, \ldots, e_{n-r-1})\oplus (e_{n+1})\oplus (e_{n+r+2}, \ldots, e_d), \]

\[ N=(e_{n-r},\ldots , e_{n-1})\oplus (e_n)\oplus (e_{n+2},\ldots , e_{n+r+1}), \]

\begin{align*} {\rm with}\quad \langle e_i , e_{d+1-j} \rangle&= \delta_{ij}, \quad \hbox{for } i< n-r \text{ or } n+r+1< i, \\ \langle e_i , e_{d+1-j} \rangle &= \pi \delta_{ij}, \quad \hbox{for } n-r\leq i\leq n+r+1,\ i\neq n,\ i\neq n+1, \\ \langle e_n , e_n \rangle &= \pi, \quad \langle e_{n+1} , e_{n+1} \rangle = 1, \quad \langle e_n , e_{n+1} \rangle = 0. \end{align*}

(3) $d= 2n+1$, $\delta (\Lambda )=2r$:

\[ {\Lambda}_{{\mathcal{O}}_{F'}}=(e_1, \ldots, e_{n-r})\oplus (e_{n-r+1},\ldots , e_{n})\oplus (e_{n+1})\oplus (e_{n+2},\ldots , e_{n+r+1})\oplus (e_{n+r+2}, \ldots, e_d), \]

\[ M=(e_1, \ldots, e_{n-r})\oplus (e_{n+1})\oplus (e_{n+r+2}, \ldots, e_d), \]

\[ N=(e_{n-r+1},\ldots , e_{n})\oplus (e_{n+2},\ldots , e_{n+r+1}), \]

\begin{align*} {\rm with}\quad \langle e_i , e_{d+1-j} \rangle &= \delta_{ij}, \quad \hbox{for } i< n+1-r \text{ or } i=n+1 \text{ or } n+1+r< i, \\ \langle e_i , e_{d+1-j} \rangle &= \pi \delta_{ij}, \quad \hbox{for } n+1-r\leq i\leq n+1+r, \text{ and } i\neq n+1. \end{align*}

(4) $d= 2n+1$, $\delta (\Lambda )=2r+1$:

\begin{gather*} {\Lambda}_{{\mathcal{O}}_{F'}}=(e_1, \ldots, e_{n-r})\oplus (e_{n-r+1},\ldots , e_{n+r+1})\oplus (e_{n+r+2}, \ldots, e_d), \end{gather*}

\begin{gather*} M=(e_1, \ldots, e_{n-r})\oplus (e_{n+r+2}, \ldots, e_d),\quad N=(e_{n-r+1},\ldots , e_{n+r+1}), \end{gather*}

\begin{gather*} {\rm with}\quad \langle e_i , e_{d+1-j} \rangle = \delta_{ij}, \quad \text{for } i< n+1-r \text{ or } n+1+r < i, \\ \langle e_i , e_{d+1-j}\rangle = \pi \delta_{ij}, \quad \text{for } n+1-r\leq i\leq n+1+r . \end{gather*}

In all the above, the parentheses give a short-hand notation for the ${\mathcal {O}}_{F'}$-lattice generated by the vectors included. For simplicity, we omit the notation of the base change of $M$, $N$. In all cases, we will denote by $S$ the (symmetric) matrix with entries $\langle e_i, e_j\rangle$ where $\{e_i\}$ is the basis above. We can then write

\[ S=S_1+\pi S_2 \]

where $S_1$, $S_2$ both have entries only $0$ or $1$. For example, in case (1) we write

\[ S_1 := \begin{pmatrix} & & 1^{(n-r)} \\ & 0^{(2r)} & \\ 1^{(n-r)} & & \\ \end{pmatrix},\quad S_2:=\begin{pmatrix} & & 0^{(n-r)} \\ & 1^{(2r)} & \\ 0^{(n-r)} & & \\ \end{pmatrix}. \]

2.3 Spinor groups

The Clifford algebra of $V$ is a ${\mathbb {Z}}/2{\mathbb {Z}}$-graded $F$-algebra denoted by

\[ C(V)=C^+(V) \oplus C^- (V). \]

It is a vector space of rank $2^d$ over $F$, generated as an algebra by the image of a canonical injection $V\hookrightarrow C^-(V)$ satisfying $v\cdot v=Q(v)$. The canonical involution on $C(V)$ is the $F$-linear endomorphism $c\mapsto c^*$ characterized by $(v_1 \cdots v_m)^*=v_m\cdots v_1,$ for $v_1,\ldots, v_m\in V$.

For an $F$-algebra $R$, the tensor product $V_R=V\otimes _{F} R$ is a non-degenerate quadratic space over $R$ with Clifford algebra $C(V_R)=C(V)\otimes _{F}R$. The spinor similitude group $G={\rm GSpin}(V)$ is the reductive group over $F$ with $R$-points

\[ {\rm GSpin}(V)(R) = \{ g\in C^+(V_R)^\times : g V_R g^{-1} =V_R,\ g^*g \in R^\times \}, \]

and the spinor similitude $\eta :{\rm GSpin}(V) \to {{\mathbb {G}}_{\rm m}}$ is the character $\eta (g) = g^* g$.

The conjugation action of $G$ on $C(V)$ leaves invariant the $F$-submodule $V$, and this action of $G$ on $V$ is denoted $g\cdot v = gvg^{-1}$. There is a short exact sequence of reductive group schemes

(2.3.1)\begin{equation} 1 \to {{\mathbb{G}}_{\rm m}} \to G \xrightarrow{g\mapsto g \cdot} {\rm SO}(V) \to 1 \end{equation}

over $F$, and the restriction of $\eta$ to the central ${{\mathbb {G}}_{\rm m}}$ is $z\mapsto z^2$.

The spinor group ${\rm Spin}(V)$ is the kernel of $\eta$ and there is a short exact sequence of reductive groups

(2.3.2)\begin{equation} 1\to {\rm Spin}(V)\to G={\rm GSpin}(V)\xrightarrow{\eta} {{\mathbb{G}}_{\rm m}}\to 1. \end{equation}

We have $G_{\rm der}={\rm Spin}(V)$ which is simply connected. Hence, $\pi _1(G)=\pi _1({{\mathbb {G}}_{\rm m}})\simeq {\mathbb {Z}}$ with trivial Galois action. In particular, we have $\pi _1(G)_I\simeq {\mathbb {Z}}$ for the coinvariants of the action by the inertia group $I={\rm Gal}(\bar F/\breve F)$. Recall that $\pi _1(G)_I$ is the target of the Kottwitz homomorphism

\[ \kappa: G(\breve F)\to \pi_1(G)_{I}. \]

Hence, in our case, the Kottwitz homomorphism is

\[ \kappa: {\rm GSpin}(V)(\breve F)\to {\mathbb{Z}}. \]

In fact, by the definition of $\kappa$ and (2.3.2), we can see that

(2.3.3)\begin{equation} \kappa(g)={\rm val}(\eta(g)), \end{equation}

that is, $\kappa$ is given by the valuation of the spinor similitude.

2.4 Orthogonal parahoric groups

We first consider the special orthogonal group ${\rm SO}(V)$. The self-dual lattice chain

\[ \cdots\subset \pi{\Lambda}^\vee\subset {\Lambda}\subset {\Lambda}^\vee\subset \pi^{-1}{\Lambda}\subset\cdots \]

gives a point $x_{\Lambda }$ of the Bruhat–Tits building ${\mathcal {B}}({\rm SO}(V), F)$ (see, for example, [Reference Bruhat and TitsBT87]). Let us consider the subgroup

\[ K=K_{\Lambda}=\{g\in {\rm SO}(V)\mid g(\Lambda)=\Lambda\} \subset {\rm SO}(V) \]

which preserves the lattice. Then every element $g\in K$ also preserves the dual lattice, $g({\Lambda }^\vee )={\Lambda }^\vee$, and so it induces elements $\bar g_1\in {\rm GL}(\Lambda /\pi \Lambda ^\vee )$, $\bar g_2\in {\rm GL}(\Lambda ^\vee /\Lambda )$, which lie in the corresponding orthogonal groups for the forms $\langle \,{,}\,\rangle _1$, $\langle \,{,}\,\rangle _2$. We can see that $g\mapsto (\bar g_1, \bar g_2)$ gives a group homomorphism

\[ K\to {\rm O}(\Lambda/\pi\Lambda^\vee)\times {\rm O}(\Lambda^\vee/\Lambda). \]

Composing with

\[ \det\times \det: {\rm O}(\Lambda/\pi\Lambda^\vee)\times {\rm O}(\Lambda^\vee/\Lambda)\to \{\pm 1\}\times \{\pm 1\} \]

gives $\varepsilon : K\to \{\pm 1\}\times \{\pm 1\}$. The corresponding parahoric subgroup $K^\circ \subset K$ is the kernel of $\varepsilon$. (See [Reference Bruhat and TitsBT87] and [Reference TitsTit79, Example 3.12]. The map $\varepsilon$ can also be related to the Kottwitz homomorphism for ${\rm SO}(V)$ which is given by the spinor norm.)

Set $i: \Lambda \to \Lambda ^\vee$, $j: \pi \Lambda ^\vee \to \Lambda$ for the natural ${\mathcal {O}}$-linear inclusions. We have $i\cdot j=\pi$.

Let $R$ be an ${\mathcal {O}}$-algebra. For simplicity, set ${\Lambda }_R=\Lambda \otimes _{\mathcal {O}} R$, ${\Lambda }^\vee _R={\Lambda }^\vee \otimes _{\mathcal {O}} R$. We identify

\[ {\Lambda}^\vee_R={\rm Hom}_R({\Lambda}_R, R) \]

using the form $\langle \,{,}\,\rangle _R=\langle \,{,}\,\rangle \otimes _{\mathcal {O}} R$. Consider the group scheme ${\mathcal {G}}={\mathcal {G}}_{\Lambda }$ over ${\rm Spec}\,({\mathcal {O}})$ which has $R$-valued points given by $g \in {\rm GL}(\Lambda _R)$ for which

(2.4.1)\begin{equation} \begin{matrix} \pi\Lambda^\vee_R &\xrightarrow{j_R} &\Lambda_R&\xrightarrow{i_R} & \Lambda^\vee_R\\ \pi g^\vee\downarrow\ \ \ \ && g\downarrow &&\ \ \ \downarrow g^\vee\\ \pi\Lambda^\vee_R &\xrightarrow{j_R} &\Lambda_R&\xrightarrow{i_R} & \Lambda^\vee_R \end{matrix} \end{equation}

commutes, and with $\det (g)=1$, $\det (g^\vee )=1$. In the above, $g^\vee$ denotes the $R$-linear map which is $R$-dual to $g: {\Lambda }_R\to {\Lambda }_R$.

As we can see, using Appendix [Reference Rapoport and ZinkRZ96], the group ${\mathcal {G}}$ is smooth and has ${\mathcal {O}}$-points given by $K$. The homomorphism $\varepsilon$ above is the composition

\[ K={\mathcal{G}}({\mathcal{O}})\to {\mathcal{G}}(k)\to \{\pm 1\}\times \{\pm 1\} \]

and the kernel gives the neutral component ${\mathcal {G}}^\circ$ of ${\mathcal {G}}$. As in [Reference Bruhat and TitsBT87], ${\mathcal {G}}={\mathcal {G}}_{\Lambda }$ and ${\mathcal {G}}^\circ ={\mathcal {G}}^\circ _{\Lambda }$, are the Bruhat–Tits group schemes that correspond to $x_{\Lambda }$, or, in other words, to $K$ and $K^\circ$, respectively.

2.5 Spinor parahoric groups

Consider the (extended) Bruhat–Tits building ${\mathcal {B}}^e(G, F)$ for $G={\rm GSpin}(V)$ over $F$. The central exact sequence (2.3.1) induces by [Reference Bruhat and TitsBT84, (4.2.15)], or [Reference LandvogtLan00, Theorem 2.1.8], a canonical $G(F)$-equivariant map

\[ {\mathcal{B}}^e(G, F)\to {\mathcal{B}}^e({\rm SO}(V), F)={\mathcal{B}}({\rm SO}(V), F) \]

which lifts an identification

\[ {\mathcal{B}}(G, F)\xrightarrow{\sim} {\mathcal{B}}({\rm SO}(V), F) \]

between the (classical) Bruhat–Tits buildings. Consider now $x\in {\mathcal {B}}^e(G, F)$ which we assume maps to the point $x_{\Lambda }\in {\mathcal {B}}(G, F)\simeq {\mathcal {B}}({\rm SO}(V), F)$ defined by a vertex lattice ${\Lambda }\subset V$. Then, the stabilizer of $x$ is the subgroup

\[ G(F)_x=\{g\in {\rm GSpin}(V)(F)\mid g\Lambda g^{-1}=\Lambda,\ \eta(g)\in {\mathcal{O}}^\times\}. \]

By (2.3.3), $G(\breve F)_x\subset {\rm ker}(\kappa )$. Hence, by [Reference Pappas and RapoportPR08, Appendix by Haines and Rapoport], the corresponding Bruhat–Tits group scheme ${\mathcal {G}}_x$ is connected (i.e. ${\mathcal {G}}_x={\mathcal {G}}^\circ _x$), and it is the corresponding parahoric group scheme over ${\mathcal {O}}$ for ${\rm GSpin}(V)$ over $F$. By [Reference Kisin and PappasKP18, Proposition 1.1.4], we have an exact sequence

(2.5.1)\begin{equation} 1\to {{\mathbb{G}}_{\rm m}}\to {\mathcal{G}}_x \to {\mathcal{G}}_\Lambda^\circ\to 1 \end{equation}

of group schemes over ${\rm Spec}\,({\mathcal {O}})$ which extends (2.3.1).

Remark 2.5.2 Assume that we are in one of the cases (1)–(4) listed in § 2.2; in particular, we assume $F=F'$ in the notation of § 2.2. Using the identification ${\mathcal {B}}({\rm GSpin}(V), F)\xrightarrow {\sim } {\mathcal {B}}({\rm SO}(V), F)$ and the description of the building for ${\mathcal {B}}({\rm SO}(V), F)$ given in [Reference Bruhat and TitsBT87], we see that each maximal compact subgroup of ${\rm GSpin}(V)$ is of the form $G(F)_x$ for $x=x_{\Lambda }$ given, as above, by some vertex lattice ${\Lambda }$ with $\delta _*({\Lambda })\neq 2$. As we mentioned above, when $\delta _*({\Lambda })=0$, $G(F)_x$ is hyperspecial. When $\delta _*({\Lambda })=1$, $G(F)_x$ is special. The vertex lattices ${\Lambda }$ with $\delta _*({\Lambda })=2$ (i.e. $\delta ({\Lambda })=2$ or $\delta ({\Lambda })=d-2$), give parahoric subgroups which are not maximal. Indeed, when $\delta ({\Lambda })=2$, there are exactly two self-dual lattices $\Lambda _0$, $\Lambda '_0$, that fit in an oriflamme configuration:

Then $G(F)_x$ is the intersection of the two hyperspecial subgroups ${\rm GSpin}({\Lambda }_0)$, ${\rm GSpin}({\Lambda }'_0)$. If $\delta ({\Lambda })=d-2$, then there are exactly two lattices ${\Lambda }_1$ and ${\Lambda }_1'$ with ${\Lambda }_1^\vee =\pi {\Lambda }_1$, ${{\Lambda }'}_1^\vee =\pi {\Lambda }'_1$, that fit in a similar oriflamme configuration between ${\Lambda }^\vee$ and $\pi ^{-1}{\Lambda }$. The group $G(F)_x$ is again the intersection of the two hyperspecial subgroups which stabilize ${\Lambda }_1$ and ${\Lambda }_1'$.

3.1 Quadrics

Suppose that $R$ is an ${\mathcal {O}}$-algebra and that $(W, \langle \,{,}\,\rangle )$ is a free $R$-module of rank $d$ with a symmetric $R$-bilinear form

\[ \langle\,{,}\,\rangle: W\times W\to R. \]

We denote by ${\mathcal {Q}}(W, \langle \,{,}\,\rangle )$, or simply by ${\mathcal {Q}}(W)\subset {\mathbb {P}}(W)\simeq {\mathbb {P}}^{d-1}_R$ when the form is understood, the closed subscheme parametrizing isotropic $R$-lines $L$ (i.e. locally $R$-free direct summands of $W$ of rank $1$) with $\langle L,L\rangle =0$.

3.2 Linked quadrics

Suppose now that $V$ and ${\Lambda }\subset V$ are as in § 2.1. We assume

\[ \delta={\rm length}_{\mathcal{O}}({\Lambda}^\vee/{\Lambda})\leq d/2. \]

Consider the ${\mathcal {O}}$-scheme $P({\Lambda }, {\Lambda }^\vee )$ parametrizing linked isotropic lines

\[ (L, L')\in {\mathcal{Q}}(\Lambda)\times_{\mathcal{O}} {\mathcal{Q}}(\Lambda^\vee)\subset {\mathbb{P}}(\Lambda)\times_{\mathcal{O}} {\mathbb{P}}(\Lambda^\vee) \]

where the ‘linked’ condition for an $R$-valued point $(L, L')$ is that

\[ j_R(\pi L')\subset L,\quad i_R(L)\subset L', \]

that is, we have a commutative diagram

\[ \begin{matrix} \pi{\Lambda}^\vee_R & \xrightarrow{j_R} & {\Lambda}_R & \xrightarrow{i_R} & {\Lambda}^\vee_R\\ \cup & & \cup & & \cup \\ \pi L' & \xrightarrow{ \ } & L & \xrightarrow{ \ } & L'. \end{matrix} \]

(Of course, isotropic means that we require $\langle L, L\rangle _R=0$, $\pi \langle L', L' \rangle _R =0$.) There is a natural isomorphism

\[ P({\Lambda}, {\Lambda}^\vee)\otimes_{\mathcal{O}} F\cong {\mathcal{Q}}(V), \]

since, for an $F$-algebra $R$, we have ${\Lambda }_R={\Lambda }^\vee _R=V_R$.

We set ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )$ for the (flat) reduced closure of the generic fiber ${\mathcal {Q}}(V)$ in the ${\mathcal {O}}$-scheme $P({\Lambda }, {\Lambda }^\vee )$. By (2.4.1), the group scheme ${\mathcal {G}}$ acts on $P(\Lambda, {\Lambda }^\vee )$ and also on the closure ${\mathcal {Q}}(\Lambda,\Lambda ^\vee )$. If $\delta =0$, then it is easy to see that ${\mathcal {Q}}({\Lambda },{\Lambda }^\vee )={\mathcal {Q}}({\Lambda })$ is a smooth quadric hypersurface in ${\mathbb {P}}^{d-1}_{\mathcal {O}}$. Part (A) of the following theorem was suggested to the authors by B. Howard.

Theorem 3.2.1 Assume $\delta >0$.

(A) The $R$-valued points of the scheme ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )$ are in bijection with the set of pairs $(L, L')$ of $R$-lines $L\subset \Lambda _R$, $L'\subset \Lambda ^\vee _R$, such that $j_R(\pi L')\subset L$, $i_R(L)\subset L',$ and

\[ \langle L, L'\rangle_R=0, \]

for the perfect $R$-bilinear pairing $\langle \,{,}\,\rangle _R: \Lambda _R\times \Lambda ^\vee _R\to R$ induced by $\langle \,{,}\,\rangle$.

(B) The scheme ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )$ is normal and is a relative local complete intersection, flat and projective over ${\rm Spec}\,({\mathcal {O}})$. Its generic fiber is the smooth quadric ${\mathcal {Q}}(V)$ in ${\mathbb {P}}(V)$. The scheme is regular if and only if $\delta = 1$.

(C) The reduced special fiber of ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )$ is the union of three reduced subschemes $Z_0$, $Z_1$ and $Z_2$ of ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )_k$, defined as follows.

• – $Z_0$ is the locus of $(L, L')$ for which $i_R(L)=0$ and $j_R(\pi L')=0$.

• – $Z_1$ is the locus of $(L, L')$ for which $L'\subset \Lambda ^\vee _R=(\Lambda ^\vee /\pi \Lambda ^\vee )\otimes _kR$ lies in the $R$-submodule $(\Lambda /\pi \Lambda ^\vee )_R\subset \Lambda ^\vee _R$, and is such that $\langle L' ,\, L' \rangle _{1, R}=0$.

• – $Z_2$ is the locus of $(L, L')$ for which $\pi ^{-1}L\subset (\pi ^{-1}\Lambda )_R=(\pi ^{-1}\Lambda / \Lambda )\otimes _kR$ lies in the $R$-submodule $( \Lambda ^\vee / \Lambda )_R\subset (\pi ^{-1}\Lambda )_R$, and is such that $\langle \pi ^{-1} L ,\, \pi ^{-1}L \rangle _{2, R}=0$.

These have the following properties.

1. (1) The subschemes $Z_1$, $Z_2$ are projective space bundles over the quadric hypersurfaces ${\mathcal {Q}}(\Lambda /\pi \Lambda ^\vee, \langle \,{,}\,\rangle _1)\subset {\mathbb {P}}(\Lambda /\pi \Lambda ^\vee )$ and ${\mathcal {Q}}(\Lambda ^\vee /\Lambda, \langle \,{,}\,\rangle _2)\subset {\mathbb {P}}(\Lambda ^\vee /\Lambda )$, respectively.

2. (2) $Z_0\simeq {\mathbb {P}}(\Lambda ^\vee /\Lambda )\times {\mathbb {P}}(\Lambda /\pi \Lambda ^\vee )$ and we have

   \[ Z_0\cap Z_1\simeq {\mathcal{Q}}(\Lambda/\pi\Lambda^\vee, \langle\,{,}\,\rangle_1)\times {\mathbb{P}}(\Lambda^\vee/\Lambda), \]
   \[ Z_0\cap Z_2\simeq {\mathbb{P}}(\Lambda/\pi\Lambda^\vee)\times {\mathcal{Q}}(\Lambda^\vee/\Lambda, \langle\,{,}\,\rangle_2), \]
   for the scheme theoretic intersections.

3. (3) div $(\pi )=2\cdot (Z_0)+(Z_1)+(Z_2)$ as Weil divisors on ${\mathcal {Q}}({\Lambda }, {\Lambda }^\vee )$.

If $\delta >2$, then $Z_0$, $Z_1$ and $Z_2$ are smooth and irreducible. If $\delta =2$, then $Z_0$, $Z_1$ are smooth and irreducible but $Z_2$ is either irreducible or the disjoint union of two smooth irreducible components. Finally, if $\delta =1$, then ${\mathcal {Q}}(\Lambda ^\vee /\Lambda, \langle \,{,}\,\rangle _2)$ and $Z_2$ are empty and $Z_0$, $Z_1$ are smooth and irreducible.

Proof. Write

\[ \Lambda=M\oplus N,\quad \Lambda^\vee=M\oplus \pi^{-1}N \]

with $M$, $N$ free ${\mathcal {O}}$-submodules of ${\Lambda }$ such that $\langle M, N\rangle =0$ and with the form $\langle \,{,}\,\rangle$ perfect on $M$ and $\pi ^{-1}\langle \,{,}\,\rangle$ perfect on $N$. We have ${\rm rank}_{\mathcal {O}}(N)=\delta$, ${\rm rank}_{\mathcal {O}}(M)=d-\delta$. Note that

\[ \bar M:=M\otimes_{\mathcal{O}} k={\Lambda}/\pi{\Lambda}^\vee, \quad \bar N:=N\otimes_{\mathcal{O}} k\simeq {\Lambda}^\vee/{\Lambda}. \]

Set

\[ \langle\,{,}\,\rangle_1 : M\times M\to {\mathcal{O}},\quad \langle m, m' \rangle_1=\langle m, m'\rangle, \]

\[ \langle\,{,}\,\rangle_2 : N\times N\to {\mathcal{O}},\quad \langle n, n' \rangle_2=\pi^{-1}\langle n, n'\rangle. \]

Now write

\[ L=(f)=(m+n),\quad L'=(f')=(m'+\pi^{-1}n') \]

with $m$, $m'\in M_R$ and $n$, $n'\in N_R$. The linking conditions are

\[ m+n=u(m'+\pi^{-1}n'),\quad \pi m'+n'=\pi(m'+\pi^{-1}n')=v(m+n), \]

for some $u$, $v\in R$. These give $m=u m'$, $n'=vn$, $uv=\pi$.

The isotropic conditions are

\[ \langle f, f\rangle=\langle m, m\rangle_1+\pi\langle n , n\rangle_2=0,\quad \pi\langle f', f'\rangle =\pi\langle m', m'\rangle_1+\langle n', n'\rangle_2=0. \]

These translate respectively to

\[ u^2\langle m', m'\rangle_1+uv\langle n , n\rangle_2=0,\quad uv\langle m', m'\rangle_1+v^2\langle n, n\rangle_2=0. \]

In the flat closure, they amount to a single equation:

(3.2.2)\begin{equation} u\langle m', m'\rangle_1+v\langle n , n\rangle_2=0. \end{equation}

By the above, we have $L=(um'+n)$, $L'=(m'+\pi ^{-1}vn)$, and so (3.2.2) amounts to

\[ \langle L, L'\rangle_R=0. \]

Denote by ${\mathcal {Q}}^\sharp (\Lambda, \Lambda ^\vee )$ the closed subscheme of ${\mathbb {P}}(\Lambda )\times _{\mathcal {O}} {\mathbb {P}}(\Lambda ')$ given by pairs $(L, L')$ such that $j_R(\pi L')\subset L$, $i_R(L)\subset L',$ and $\langle L, L'\rangle _R=0$. By the above discussion, the flat closure ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )$ is a closed subscheme of ${\mathcal {Q}}^\sharp (\Lambda, \Lambda ^\vee )$ with the same generic fiber. To show (A), it is enough to show that ${\mathcal {Q}}^\sharp (\Lambda, \Lambda ^\vee )$ is flat over ${\mathcal {O}}$.

Consider the affine scheme over ${\rm Spec}\,({\mathcal {O}})$ given as the product

\[ {\mathcal{X}}:= {\mathbb{A}}(M)\times_{\mathcal{O}} {\mathbb{A}}(N)\times_{\mathcal{O}} {\rm Spec}\,({\mathcal{O}}[u, v]/(uv-\pi))={\rm Spec}\,({\mathcal{O}}[\underline x, \underline y, u, v]/(uv-\pi)). \]

(Here we set ${\mathbb {A}}(M)={\rm Spec}\,({\rm Sym}^\bullet _{\mathcal {O}}(M^\vee ))$, and similarly for ${\mathbb {A}}(N)$. We identify ${\mathbb {A}}(M)$, ${\mathbb {A}}(N)$ with ${\rm Spec}\,({\mathcal {O}}[\underline x])$, ${\rm Spec}\,({\mathcal {O}}[\underline y])$, after picking a basis of $M$, $N$, respectively.) Let ${{\mathcal {X}}^0={\mathcal {X}}- (T_1\cup T_2\cup T_0)}$ be the open subscheme of ${\mathcal {X}}$ which is the complement of the union of the three subschemes $T_1$, $T_2$, $T_0$, defined respectively by the ideals

\[ (\underline x, v),\quad (\underline y, u), \quad (\underline x, \underline y). \]

The scheme ${\mathcal {X}}^0$ supports an action of the torus ${\mathbb {G}}_m\times {\mathbb {G}}_m$ given by

\[ (\lambda, \mu)\cdot (\underline x, \underline y, u, v)=(\lambda \underline x, \mu \underline y, \lambda^{-1}\mu u, \lambda\mu^{-1} v). \]

The subscheme ${\mathcal {Y}}$ of ${\mathcal {X}}^0$ defined by the equation

(3.2.3)\begin{equation} u\cdot \langle \underline x, \underline x\rangle_1+v\cdot \langle \underline y, \underline y\rangle_2=0, \end{equation}

is stable under the torus action. The quotient

\[ {\mathcal{Y}}/({\mathbb{G}}_m\times{\mathbb{G}}_m) \]

is isomorphic to ${\mathcal {Q}}^\sharp (\Lambda, \Lambda ^\vee )$ and is flat over ${\mathcal {O}}$. This shows ${\mathcal {Q}}^\sharp (\Lambda, \Lambda ^\vee )={\mathcal {Q}}(\Lambda, \Lambda ^\vee )$ which is part (A).

Let us now discuss the subschemes $Z_0$, $Z_1$, $Z_2$, of the special fiber ${\mathcal {Q}}^\sharp (\Lambda, \Lambda ^\vee )_k={\mathcal {Q}}(\Lambda, \Lambda ^\vee )_k$. Suppose that $R$ is a $k$-algebra and assume that $(L, L')$ is a pair of $R$-lines $L\subset {\Lambda }_R$, $L'\subset {\Lambda }^\vee _R$, which are linked, that is, satisfy $i_R(L)\subset L'$ and $j_R(\pi L')\subset L$. Then $L'\subset (\Lambda /\pi \Lambda ^\vee )_R$ if and only if $j_R(\pi L')=(0)$. Similarly, $\pi ^{-1}L\subset (\Lambda ^\vee / \Lambda )_R$ if and only if $i_R(L)=(0)$. We first observe that any pair of $R$-lines $(L, L')$ with $L'\subset (\Lambda /\pi \Lambda ^\vee )_R$, $\pi ^{-1}L\subset (\Lambda ^\vee / \Lambda )_R$, satisfies $i_R(L)\!=(0)$, $j_R(\pi L')\!=(0)$ and so is linked, but also has $\langle L , L' \rangle _{R}=0$ and hence gives an $R$-point of $Z_0$. Suppose we are given an $R$-line $L'\subset (\Lambda /\pi \Lambda ^\vee )_R$ which is isotropic for the form $\langle \,{,}\,\rangle _{1, R}$. Then every $R$-line $L\subset \Lambda _R$ with $i_R(L)= L'$ gives an $R$-point $(L, L')$ of $Z_1$. There is a similar construction for $R$-points of $Z_2$. Finally, observe that if $R$ is a $k$-algebra which is an integral domain, since $i_R\circ j_R=0$, at least one of $i_R(L)$, $j_R(\pi L')$, has to be $(0)$. If $i_R(L)=L'$, then $j_R(\pi L')=(0)$ and then $\langle L , L' \rangle _{R}=0$ is equivalent to $\langle L' , L' \rangle _{1, R}=0$. Similarly, if $j_R(\pi L')=L$, then $i_R(L)=(0)$ and $\langle L , L' \rangle _{R}=0$ is equivalent to $\langle \pi ^{-1}L , \pi ^{-1}L \rangle _{2, R}=0$. These considerations easily imply that the reduced special fiber of ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )_k$ is the union $Z_0\cup Z_1\cup Z_2$. We can also deduce that the schemes $Z_i$ have the descriptions given in (C1) and (C2). This will also be explained below.

To obtain more precise (scheme-)theoretic information and show the remaining statements we will use the quotient description above. In this description, the subschemes $Z_2$, $Z_1$, $Z_0$ of ${\mathcal {Q}}(\Lambda, \Lambda ^\vee )$ are given by the quotient of the closures of $u=0$, $v\neq 0$, $\langle \underline y, \underline y\rangle _2=0$, of $v=0$, $u\neq 0$, $\langle \underline x, \underline x\rangle _1=0$, and of $u=v=0$, respectively.

• – $Z_2$ is given by the quotient of the subscheme $\langle \underline y, \underline y\rangle _2=0$ in the complement

  \[ [{\mathbb{A}}(\bar M)\times_k ({\mathbb{A}}(\bar N)-\{0\})\times_k {\mathbb{A}}^1]-(0\times ({\mathbb{A}}(\bar N)-\{0\})\times 0) \]
  by ${\mathbb {G}}_m\times {\mathbb {G}}_m$. Here, $(\lambda, \mu )$ acts by
  \[ (\lambda, \mu)\cdot (\underline x, \underline y, v )=(\lambda \underline x, \mu \underline y, \lambda\mu^{-1} v). \]
  This is a projective space bundle over the ${\mathbb {G}}_m$-quotient of $\langle \underline y, \underline y\rangle _2=0$ in ${\mathbb {A}}(\bar N)-\{0\}$; this last quotient is the projective quadric hypersurface ${\mathcal {Q}}(\Lambda ^\vee /{\Lambda }, \langle, \rangle _2)$. Hence, $Z_2$ is a projective space bundle over ${\mathcal {Q}}(\Lambda ^\vee /{\Lambda }, \langle, \rangle _2)$.

• – Similarly, $Z_1$ is a projective space bundle over the projective quadric hypersurface ${\mathcal {Q}}({\Lambda }/\pi {\Lambda }^\vee, \langle \, , \rangle _1)$.

• – $Z_0$ is the quotient

  \[ ({\mathbb{A}}(\bar M)-\{0\})\times_k ({\mathbb{A}}(\bar N)-\{0\})/({\mathbb{G}}_m\times {\mathbb{G}}_m), \]
  so it is a product of projective spaces ${\mathbb {P}}(\Lambda ^\vee /\Lambda )\times {\mathbb {P}}(\Lambda /\pi \Lambda ^\vee )$.

It follows easily from the above that (C1) and (C2) hold. We now show the remaining statements, in particular that ${\mathcal {Q}}({\Lambda },{\Lambda }^\vee )$ is a normal relative local complete intersection as claimed in (B), and that the identity of Weil divisors in (C3) is true.

Consider a $\bar k$-point of ${\mathcal {Q}}({\Lambda },{\Lambda }^\vee )$ obtained from a $\bar k$-valued point of ${\mathcal {Y}}$ for which $\underline x=0$. Since ${\mathcal {Y}}\subset {\mathcal {X}}^0$, we necessarily have $v\neq 0$ and hence $u=0$. Then we also have $\underline y\neq 0$ and the point lies in $Z_2$ but not on $Z_0$. The equation (3.2.3) amounts to just $\langle \underline y, \underline y\rangle _2=0$. Since we are considering a non-zero point $\underline y$, the scheme ${\mathcal {Q}}({\Lambda }, {\Lambda }^\vee )_k$ is smooth there. The same argument works for a point for which $\underline y=0$; this lies on $Z_1$ but not on $Z_0$.

It remains to deal with points for which $\underline x\neq 0$ and $\underline y\neq 0$. Consider the corresponding open subscheme

\[ {\mathcal{Q}}^0({\Lambda}, {\Lambda}^\vee)=[({\mathbb{A}}(M)-\{0\})\times_{\mathcal{O}} ({\mathbb{A}}(N)-\{0\})\times {\rm Spec}\,({\mathcal{O}}[u, v]/(uv-\pi))]/({{\mathbb{G}}_{\rm m}}\times{{\mathbb{G}}_{\rm m}}) \]

of ${\mathcal {Q}}({\Lambda }, {\Lambda }^\vee )$. We will obtain an explicit description of ${\mathcal {Q}}^0({\Lambda },{\Lambda }^\vee )$.

Recall that we can choose bases of $M$ and $N$ that give coordinates $\underline x=(x_1,\ldots, x_{d-\delta })$ on ${\mathbb {A}}(M)$ and $\underline y=(y_1, \ldots, y_\delta )$ on ${\mathbb {A}}(N)$. Denote by $Q_1(x_1, \ldots, x_{d-\delta })$, $Q_2(y_1, \ldots, y_\delta )$ the quadratic forms given as $\langle \underline x, \underline x\rangle _1/2$ and $\langle \underline y, \underline y\rangle _2/2$.

Then ${\mathcal {Q}}^0(\Lambda, \Lambda ^\vee )$ is covered by the open affines ${\mathcal {V}}_{i,j}$ with equations

\[ uv=\pi,\quad uQ_1(x_1, \ldots , x_{d-\delta})+vQ_2(y_1, \ldots , y_\delta)=0,\quad x_i=1,\quad y_j=1. \]

For simplicity, we will just consider the case $i=1$, $j=1$, and set ${\mathcal {V}}={\mathcal {V}}_{11}$.

We first consider the case $\delta =1$. Then ${\mathcal {V}}$ is given by

\[ uv=\pi,\quad uQ_1(1, x_2, \ldots , x_{d-\delta})+v=0, \]

that is,

\[ -u^2 Q_1(1, x_2, \ldots , x_{d-\delta})=\pi. \]

In this case, ${\mathcal {V}}$ and hence ${\mathcal {Q}}({\Lambda },{\Lambda }^\vee )$ is regular and its special fiber is a divisor with normal crossings.

Assume now that $\delta \geq 2$. As also later, it helps to consider the simpler ‘basic’ scheme

\[ {\mathcal{B}}={\rm Spec}\,({\mathcal{O}}[u, v, S, T]/(uv-\pi, uS+vT). \]

There is a morphism $f: {\mathcal {V}}\to {\mathcal {B}}$ given by $S\mapsto Q_1$, $T\mapsto Q_2$. We can see that $f$ is smooth. (Recall that $Q_1$ and $Q_2$ are non-degenerate.) We will now check the desired properties for ${\mathcal {B}}$: The scheme ${\mathcal {B}}$ has relative dimension $2$ over ${\rm Spec}\,({\mathcal {O}})$ and is a relative complete intersection. The radical of $(uS+vT, uv)$ is $(uS, vT, uv)$ and so the reduced special fiber of ${\mathcal {B}}$ has three smooth irreducible components given by the prime ideals $(u, v)$, $(S, v)$, $(T, u)$. The component given by $(u, v)$ is not reduced in the special fiber; the corresponding primary ideal is $(u^2, uv, v^2, uS+vT)$. The other components are reduced. We can see that ${\mathcal {B}}$ is regular in codimension $1$ and it easily follows by Serre's criterion that ${\mathcal {B}}$ is normal.

It now follows that we have a similar picture for the special fiber of ${\mathcal {V}}$ and that ${\mathcal {V}}$ is a normal relative complete intersection of relative dimension $d-2$. The result for ${\mathcal {Q}}^0({\Lambda },{\Lambda }^\vee )$ and then also ${\mathcal {Q}}({\Lambda }, {\Lambda }^\vee )$ follows.

3.3 A blow-up

Now let us consider the blow-up

\[ \beta_i: \widetilde {\mathcal{Q}}^{(i)}({\Lambda}, {\Lambda}^\vee)\to {\mathcal{Q}}(\Lambda, \Lambda^\vee) \]

along the (reduced) subscheme $Z_i$, $i=1$, $2$. We first observe that when $\delta =1$, by Remark 3.2.4, $Z_1$ is locally principal and so $\widetilde {\mathcal {Q}}^{(1)}({\Lambda }, {\Lambda }^\vee )= {\mathcal {Q}}(\Lambda, \Lambda ^\vee )$.

Proof. By Remark 3.2.4 and the above, these statements hold when $\delta =1$. Assume $\delta \geq 2$. Let us consider the blow-up $\widetilde {\mathcal {B}}$ of

\[ {\mathcal{B}}={\rm Spec}\,({\mathcal{O}}[u, v, S, T]/(uv-\pi, uS+vT) \]

along $(S, v)$. It is a closed subscheme of

\[ \{Sy-vx=0\mid (S, T, u, v)\times (x;y)\}\subset {\mathcal{B}}\times_{\mathcal{O}} {\mathbb{P}}^1_{\mathcal{O}}, \]

where ${\mathbb {P}}^1_{\mathcal {O}}$ has projective coordinates $(x;y)$. This is covered by two open affine subschemes.

(I) $y=1$. Then $S=vx$ and we obtain $uv=\pi$, $v(ux+T)=0$. Hence, we can see that the corresponding open subscheme of the blow-up $\widetilde {\mathcal {B}}$ is cut out by the equations $uv=\pi$, $T=-ux$, in the subscheme displayed above. This open subscheme is isomorphic to

\[ {\rm Spec}\,({\mathcal{O}}[u,v,x]/(uv-\pi)). \]

(II) $x=1$. Then $v=Sy$ and we obtain: $uSy=\pi$, $uS+SyT=0$. Hence, the corresponding open subscheme of the blow-up $\widetilde {\mathcal {B}}$ is cut out by the equations $uSy=\pi$, $u+yT=0$, in the subscheme displayed above. These give $STy^2=-\pi$ and this open subscheme is isomorphic to

\[ {\rm Spec}\,({\mathcal{O}}[S,T,y]/(STy^2+\pi)). \]

Notice that the ideal $(T, u)$ becomes principal on both of these charts. On (I) we have $T=-ux$, so $(T, u)=(u)$. On (II) we have $u=-yT$, so $(T, u)=(T)$. Using the universal property of the blow-up and symmetry gives that the blow-ups of $(S, v)$ and $(T, u)$ are isomorphic.

By the proof of Theorem 3.2.1, we see that the open subscheme ${\mathcal {Q}}^0({\Lambda }, {\Lambda }^\vee )\subset {\mathcal {Q}}({\Lambda }, {\Lambda }^\vee )$ is covered by open affine subschemes ${\mathcal {V}}={\mathcal {V}}_{ij}$ that support a smooth morphism

\[ f: {\mathcal{V}}\to {\mathcal{B}}={\rm Spec}\,({\mathcal{O}}[u, v, S, T]/(uv-\pi, uS+vT);\quad S\mapsto Q_1,\ T\mapsto Q_2. \]

Note that, by the same proof, ${\mathcal {Q}}({\Lambda },{\Lambda }^\vee )$ is smooth at the points of ${\mathcal {Q}}({\Lambda },{\Lambda }^\vee )-{\mathcal {Q}}^0({\Lambda }, {\Lambda }^\vee )$ and the divisors $Z_1$ and $Z_2$ are principal at these points. Therefore, the blow-ups $q_i:\widetilde {\mathcal {Q}}^{(i)}({\Lambda }, {\Lambda }^\vee )\to {\mathcal {Q}}({\Lambda }, {\Lambda }^\vee )$ are isomorphisms locally over ${\mathcal {Q}}({\Lambda },{\Lambda }^\vee )-{\mathcal {Q}}^0({\Lambda }, {\Lambda }^\vee )$. To show our result it is enough to consider the blow-ups of ${\mathcal {V}}$ at $Z_1\cap {\mathcal {V}}$ and at $Z_2\cap {\mathcal {V}}$. Since $Z_1\cap {\mathcal {V}}$ is given by the ideal $(Q_1, u)$, the blow-up $\widetilde {\mathcal {V}}_1$ of ${\mathcal {V}}$ along $Z_1\cap {\mathcal {V}}$ is obtained as the fiber product

\[ \widetilde{\mathcal{V}}_1\simeq {\mathcal{V}}\times_{{\mathcal{B}} }\widetilde{\mathcal{B}}. \]

The morphism $\widetilde {\mathcal {V}}_1\to \widetilde {\mathcal {B}}$ is smooth and (b) and (c) now follow from our discussion above and Theorem 3.2.1 and its proof. Part (a) ($\widetilde {\mathcal {Q}}^{(1)}({\Lambda }, {\Lambda }^\vee )\simeq \widetilde {\mathcal {Q}}^{(2)}({\Lambda }, {\Lambda }^\vee )$) follows from $\widetilde {\mathcal {V}}_1\simeq \widetilde {\mathcal {V}}_2$ since, by the observation above, the blow-ups of ${\mathcal {B}}$ at $(S, v)$ and $(T, u)$ are isomorphic.

4.1 Local models after [PZ13]

We briefly recall some of the constructions in [Reference Pappas and ZhuPZ13].

Let $G$ be a connected reductive group over $F$. Assume that $G$ splits over a tamely ramified extension of $F$. Let $\{\mu \}$ be the conjugacy class of a minuscule geometric cocharacter $\mu : \mathbb {G}_{m \bar F} \rightarrow G_{\bar F}$. Let $K$ be a parahoric subgroup of $G(F)$, which is the connected stabilizer of some point $x$ in the (extended) Bruhat–Tits building $\mathcal {B}^e(G,F)$ of $G(F)$. Define $E$ to be the extension of $F$ which is the field of definition of the conjugacy class $\{\mu \}$. The construction of the local model ${\rm M}^{\rm loc}_K(G,\{\mu \})$ is done as follows.

First, give an affine group scheme $\underline {\mathcal {G}}$ which is smooth over ${\rm Spec}\,({\mathcal {O}}_F[t])$ and which, among other properties, satisfies the following conditions.

1. (i) The base change of $\underline {\mathcal {G}}$ by ${\rm Spec}\,({\mathcal {O}}_F) \rightarrow {\rm Spec}\,({\mathcal {O}}_F[t]) = \mathbb {A}^1_{{\mathcal {O}}_F}$ given by $t \rightarrow \pi$ is the Bruhat–Tits group scheme ${\mathcal {G}}$ which corresponds to $K$ (see [Reference Bruhat and TitsBT84]).

2. (ii) The group scheme $\underline {\mathcal {G}}\otimes _{{\mathcal {O}}_F [t]} {\mathcal {O}}_F [t, t^{-1}]$ is reductive.

Next, consider the global (‘Beilinson–Drinfeld’) affine Grassmannian

\[ {\rm Aff}_{\underline{\mathcal{G}}} \rightarrow \mathbb{A}^1_{{\mathcal{O}}_F} \]

given by $\underline {\mathcal {G}}$, which is an ind-projective ind-scheme. The base change $t \rightarrow \pi$ gives an equivariant isomorphism

\[ \text{Aff}_G \xrightarrow{\sim} \text{Aff}_{\underline{\mathcal{G}} }\times_{\mathbb{A}^1_{{\mathcal{O}}_F}} {\rm Spec}\,(F) \]

where $\text {Aff}_G$ is the affine Grassmannian of $G$; this is the ind-projective ind-scheme over ${\rm Spec}\,(F)$ that represents the fpqc sheaf associated to

\[ R \rightarrow G(R(\!( t)\!))/G(R[\![ t]\!]), \]

where $R$ is an $F$-algebra (see also [Reference Pappas and RapoportPR08]). The cocharacter $\mu$ gives an $\bar F(\!( t)\!)$-valued point $\mu (t)$ of $G$. This gives a $\bar F$-point $[\mu (t)]= \mu (t) G ( \bar F[\![ t]\!])$ of $\text {Aff}_G$. Since $\mu$ is minuscule and $\{\mu \}$ is defined over the reflex field $E$, the orbit

\[ G(\bar F[\![ t]\!])[\mu(t)] \subset \text{Aff}_G (\bar F) \]

is equal to the set of $\bar F$-points of a closed subvariety $X_{\mu }$ of $\text {Aff}_{G,E}= \text {Aff}_G \otimes _{F }E .$

Definition 4.1.1 The local model ${\rm M}^{\rm loc}_K(G,\{\mu \})$ is the flat projective scheme over ${\rm Spec}\,({\mathcal {O}}_E)$ with ${\mathcal {G}}\otimes _{{\mathcal {O}}_F}{\mathcal {O}}_E$-action given by the reduced Zariski closure of the image of

\[ X_{\mu} \subset \text{Aff}_{G, E} \xrightarrow{\sim} \text{Aff}_{\underline{\mathcal{G}} }\times_{\mathbb{A}^1_{{\mathcal{O}}_F}} {\rm Spec}\,(E) \]

in the ind-scheme $\text {Aff}_{\underline {\mathcal {G}}}\times _{\mathbb {A}^1_{{\mathcal {O}}_F}} {\rm Spec}\,({\mathcal {O}}_E)$.

These ‘Pappas–Zhu’ (PZ) local models of [Reference Pappas and ZhuPZ13] are independent of choices in their construction [Reference He, Pappas and RapoportHPR20, Theorem 2.7] and have the following property (see [Reference He, Pappas and RapoportHPR20, Proposition 2.14] and its proof).

Proposition 4.1.2 If $F'/F$ is a finite unramified extension, then (equivariantly)

\[ {\rm M}^{\rm loc}_K(G, \{\mu\}) \otimes_{{\mathcal{O}}_E}{\mathcal{O}}_{E'}\xrightarrow{\sim } {\rm M}^{\rm loc}_{K'}(G\otimes_FF',\{\mu\otimes_FF'\}). \]

Note that here the reflex field $E'$ of $(G\otimes _FF', \{\mu \otimes _FF'\})$ is the join of $E$ and $F'$. Also, $K'$ is the parahoric subgroup of $G\otimes _FF'$ with $K = K' \cap G$.

4.2 Lattices over $\mathcal {O}[u]$ and orthogonal local models

We now concentrate our attention on $G={\rm SO}(V)$, where $V$ is an $F$-vector space of dimension $d\geq 5$ equipped with a non-degenerate symmetric $F$-bilinear form $\langle \,{,}\,\rangle$. We consider the minuscule coweight $\mu : \mathbb {G}_m \rightarrow {\rm SO}(V)$ to be given by $\mu (t) = \text {diag} ( t^{-1}, 1,\ldots,1, t )$, defined over $E=F$. We take $K$ to be the parahoric subgroup of ${\rm SO}(V)$ which is the connected stabilizer of a vertex lattice $\Lambda \subset V$, with $\delta ({\Lambda })={\rm length}_{{\mathcal {O}}_F}(\Lambda ^\vee /\Lambda )$.

For simplicity, we set ${\mathcal {O}}={\mathcal {O}}_F$. We extend the data of the vector space $V$ with its symmetric bilinear form $\langle \,{,}\,\rangle$ from $F$ to $\mathcal {O}[u,u^{-1}]$ by following the procedure in [Reference Pappas and ZhuPZ13, 5.2, 5.3]. This is simpler to explain when we are in one of the four cases of § 2.2 (which we can assume after an unramified extension). Then we define $\mathbb {V} = \oplus ^{d}_{i=1} \mathcal {O}[u,u^{-1}] \underline {e}_i$ and let $\langle \,{,}\,\rangle : \mathbb {V} \times \mathbb {V} \rightarrow \mathcal {O}[u,u^{-1}]$ be a symmetric $\mathcal {O}[u,u^{-1}]$-bilinear form such that the value of $\langle \underline {e}_i , \underline {e}_j \rangle$ is the same as that for $V$ but with $\pi$ replaced by $u$. Similarly, we define $\underline {\mu } : \mathbb {G}_m \rightarrow {\rm SO}(\mathbb {V})$ as above by using the $\{ \underline {e}_i\}$ basis for $\mathbb {V}$. We set

\[ \mathbb{L} = \oplus_{i=1}^d \mathcal{O}[u]\cdot \underline{e}_i \subset \mathbb{V}. \]

From the above, we see that the base change of $(\mathbb {V}, \mathbb {L},\langle \,{,}\,\rangle )$ from $\mathcal {O}[u,u^{-1}]$ to $F$ given by $u \mapsto \pi$ is $(V,\Lambda,\langle \,{,}\,\rangle )$.

Let us now consider the local model ${\rm M}^{\rm loc}(\Lambda )= {\rm M}^{\rm loc}_K ({\rm SO}(V),\{\mu \})$ where $K$ is the parahoric stabilizer $K_{\Lambda }^\circ$ of $\Lambda$.

As in [Reference Pappas and ZhuPZ13], we consider the smooth, affine group scheme $\underline {\mathcal {G}}$ over $\mathcal {O}[u]$ given by $g \in {\rm SO}(\mathbb {V})$ that also preserve $\mathbb {L}$ and $\mathbb {L}^{\vee }$. The base change of $\underline {\mathcal {G}}$ by $u \mapsto \pi$ gives the Bruhat–Tits group scheme $\mathcal {G}={\mathcal {G}}_{\Lambda }$ of ${\rm SO}(V)$ which is the stabilizer of the lattice chain $\Lambda \subset \Lambda ^{\vee } \subset \pi ^{-1} \Lambda$ as in § 2.4. In this case, the global (‘Beilinson–Drinfeld’) affine Grassmannian $\text {Aff}_{\underline {\mathcal {G}}} \rightarrow {\rm Spec}\,(\mathcal {O}[u])$ represents the functor that sends an $\mathcal {O}[u]$-algebra $R$, with $u$ mapping to $r$, to the set of finitely generated projective $R[u]$-submodules $\mathcal {L}$ of

\[ \mathbb{V}\otimes_{\mathcal{O}} R[(u-r)^{-1}]=\oplus_i R[u, u^{-1}, (u-r)^{-1}] \underline e_i, \]

which are such that

\[ (u-r)^{N}\mathbb{L }_R\subset \mathcal{L} \subset (u-r)^{-N}\mathbb{L}_R \]

for some $N >0$ with ${\mathcal {L}}/(u-r)^N{\mathbb {L}}_R$, $(u-r)^{-N}\mathbb {L}_R/{\mathcal {L}}$ both $R$-projective, and which satisfy

\[ \mathcal{L}\buildrel{\delta}\over\subset \mathcal{L}^{\vee}\buildrel{d-\delta}\over\subset u^{-1}{\mathcal{L}} \]

with all successive quotients $R$-finite projective of the indicated rank. Here, we set ${\mathbb {L}}_R={\mathbb {L}}\otimes _{\mathcal {O}} R$.

As in [Reference He, Pappas and RapoportHPR20, 12.7.2] or [Reference ZachosZac20, 3.b], we see that the local modelFootnote ¹ ${\rm M}^{\rm loc}={\rm M}^{\rm loc}({\Lambda })$ is a subfunctor of ${\rm {Aff}}_{\underline {\mathcal {G}}}\otimes _{{\mathcal {O}}[u]}{\mathcal {O}}$ (with the base change given by $u\mapsto \pi$).

Proposition 4.2.2 The functor ${\rm M}^{\rm naive}({\Lambda })$ is represented by a closed projective subscheme ${\rm M}^{\rm naive}({\Lambda })$ of ${\rm {Aff}}_{\underline {\mathcal {G}}}\otimes _{{\mathcal {O}}[u]}{\mathcal {O}}$ with ${\mathcal {G}}$-action. There is a ${\mathcal {G}}$-equivariant closed immersion

\[ i: {\rm M}^{\rm loc}({\Lambda})\hookrightarrow {\rm M}^{\rm naive}({\Lambda}). \]

4.3 Spinor local models

Recall the constructions and considerations of §§ 2.3 and 2.5. In particular, set $G={\rm GSpin}(V)$ and recall the central extension

\[ 1\to {{\mathbb{G}}_{\rm m}}\to {\rm GSpin}(V)\xrightarrow{\alpha} {\rm SO}(V)\to 1. \]

Define the cocharacter $\widetilde \mu :{{\mathbb {G}}_{\rm m}}\to G$ by

\[ \mu(t)=t^{-1}f_1f_{d}+f_{d}f_1 \]

where $f_1$, $f_d$ are part of a basis $(f_i)_i$ of $V$ such that $\langle f_1, f_d \rangle =1$, $\langle f_1, f_1\rangle =\langle f_d, f_d\rangle =0$, and the arithmetic on the right-hand side takes place in the Clifford algebra $C(V)$. Under the representation $\alpha : G \to {\rm SO}(V)$, we have

\[ (\alpha\cdot \widetilde\mu(t)) \cdot f_i = \widetilde\mu(t) f_i\widetilde\mu(t)^{-1}=\begin{cases} t^{-1} f_i & \mbox{if }i=1 \\ f_i & \mbox{if } 2 \le i < d \\ t f_i & \mbox{if }i=d. \end{cases} \]

Hence, $\alpha \cdot \widetilde \mu : {{\mathbb {G}}_{\rm m}}\to {\rm SO}(V)$ is given by the minuscule $\mu (t)={\rm diag}(t^{-1}, 1, \ldots, 1, t)$.

Now let $x\in {\mathcal {B}}^e(G, F)$ be a point in the extended Bruhat–Tits building of $G(F)$ such that $x$ maps to $x_{\Lambda }$ under $\alpha _*: {\mathcal {B}}^e(G, F)\to {\mathcal {B}}({\rm SO}(V), F)$, where ${\Lambda }$ is a vertex lattice of $V$. Then the corresponding parahoric group of $G$ is given by

\[ K={\mathcal{G}}_x({\mathcal{O}})=\{g\in {\rm GSpin}(V)(F)\mid g\Lambda g^{-1}= \Lambda,\ \eta(g)\in {\mathcal{O}}^\times\}. \]

By Remark 4.1.3 and [Reference He, Pappas and RapoportHPR20, Proposition 12.4], there is an equivariant isomorphism between local models

(4.3.1)\begin{equation} {\rm M}^{\rm loc}_{K}(G, \{\widetilde\mu\})\xrightarrow{\sim} {\rm M}^{\rm loc}_{K^\circ_{\Lambda}}({\rm SO}(V), \{\mu\})={\rm M}^{\rm loc}({\Lambda}) \end{equation}

where $K^\circ _{\Lambda }$ is the parahoric stabilizer of ${\Lambda }$ in ${\rm SO}(V)$.

The equivariance here is meant in the following sense. The natural action of ${\mathcal {G}}_x={\mathcal {G}}^\circ _x$ on ${\rm M}^{\rm loc}_{K}(G, \{\widetilde \mu \})$ factors via the quotient ${\mathcal {G}}_x/{{\mathbb {G}}_{\rm m}}\simeq {\mathcal {G}}^\circ _{\Lambda }$ (see (2.5.1)) and, under the isomorphism (4.3.1), agrees with the action of the corresponding parahoric group scheme ${\mathcal {G}}^\circ _{\Lambda }$ of ${\rm SO}(V)$ on ${\rm M}^{\rm loc}({\Lambda })$.

Hence, all our results on ${\rm M}^{\rm loc}({\Lambda })$, are really also about local models for the spin similitude groups.

5.1 An open affine of the local model following [Zac20]

We continue with the same notation (see especially § 4.2). Our goal is to describe explicitly certain open affine subschemes $U^{\rm naive}$ and $U$ of ${\rm M}^{\rm naive}({\Lambda })$ and ${\rm M}^{\rm loc}({\Lambda })$, respectively. We will work over $\breve {\mathcal {O}}$ but, for simplicity, sometimes omit the base change from the notation.

We start with $U^{\rm naive}$. For an $R$-valued point ${\mathcal {L}}$ of ${\rm M}^{\rm naive}={\rm M}^{\rm naive}({\Lambda })$ we set

\[ \bar{\mathcal{L}}:={\rm Im}({\mathcal{L}})\subset {(u-\pi)^{-1}{\mathbb{L}}_R}/{(u-\pi){\mathbb{L}}_R} \]

which determines ${\mathcal {L}}$ uniquely.Footnote ²

Recall we are fixing a basis

\[ {\mathbb{L}}=\oplus_{i=1}^d\breve{\mathcal{O}}[u]\cdot \underline e_i \]

which corresponds to the basis $(e_i)$ of $\Lambda ={\mathbb {L}}/(u-\pi ){\mathbb {L}}$. This, in turn, gives an $R$-basis $\{\underline e_i, (u-\pi )^{-1}\underline e_i\}$ of $(u-\pi )^{-1}{\mathbb {L}}_R/{\mathbb {L}}_R$, for any ${\mathcal {O}}$-algebra $R$. We consider the open subscheme $U^{\rm naive}$ of ${\rm M}^{\rm naive}$ where $\bar {\mathcal {L}}$ projects isomorphically onto the quotient obtained by annihilating the second part of the basis (i.e. $\{(u-\pi )^{-1}\underline e_i\}$). We will now describe $U^{\rm naive}$ explicitly.

We set

\[ \bar{\mathcal{L}}=\bigg\{v +X(u-\pi)^{-1} v\biggm\vert v=\sum^d_{i=1} v_i \underline e_i\bigg\}, \]

where $X$ is a variable $d\times d$ matrix with entries in $R$. This naturally produces an $R$-basis $\underline e_i+X(u-\pi )^{-1}\underline e_i$ for $\bar {\mathcal {L}}$, that is, an $R$-module isomorphism $R^d\simeq \bar {\mathcal {L}}$. Then the map

\[ \Phi: {\mathcal{L}}/(u-\pi){\mathbb{L}}_R=\bar{\mathcal{L}}\hookrightarrow (u-\pi)^{-1}{\mathbb{L}}_R/(u-\pi){\mathbb{L}}_R\to (u-\pi)^{-1}{\mathbb{L}}_R/{\mathbb{L}}_R \]

of Definition 4.2.1(v) is given by the matrix $X$.

Similarly, we can write

\[ \bar{\mathcal{L}}^\vee=\bigg\{v +Y(u-\pi)^{-1} v\biggm\vert v=\sum^d_{i=1} v_i \underline e^\vee_i\bigg\}, \]

where $Y$ is a variable $d\times d$ matrix with entries in $R$. As above, the map

\[ \Psi: {\mathcal{L}}^\vee/(u-\pi){\mathbb{L}}^\vee_R = \bar{\mathcal{L}}^\vee \hookrightarrow (u-\pi)^{-1}{\mathbb{L}}^\vee_R/(u-\pi){\mathbb{L}}^\vee_R\to (u-\pi)^{-1}{\mathbb{L}}^\vee_R/{\mathbb{L}}^\vee_R \]

of Definition 4.2.1(vi) is given by the matrix $Y$.

(a) The condition that ${\mathcal {L}}^\vee$ is the dual of ${\mathcal {L}}$ gives that we have $\langle \bar {\mathcal {L}}, \bar {\mathcal {L}}^\vee \rangle =0$ under the $R$-base change of the pairing

\[ \langle\,{,}\,\rangle: (u-\pi)^{-1}{\mathbb{L}}/(u-\pi){\mathbb{L}}\times (u-\pi)^{-1}{\mathbb{L}}^\vee/(u-\pi){\mathbb{L}}^\vee\to (u-\pi)^{-2} \breve{\mathcal{O}}[u]/\breve{\mathcal{O}}[u]. \]

Hence, on this affine chart

\[ \langle v+X(u-\pi)^{-1} v, w+Y(u-\pi)^{-1} w\rangle=0. \]

This is equivalent to

(5.1.1)\begin{equation} Y+X^t=0,\quad X^t\cdot Y=0 \end{equation}

since $\langle Xv, w\rangle =\langle w, X^tw\rangle$.

(b) The conditions $\wedge ^2\Phi =0$, $\wedge ^2\Psi =0$, immediately translate to

(5.1.2)\begin{equation} \wedge^{2}\!\!X=0,\quad \wedge^2 Y=0. \end{equation}

(c) As in [Reference ZachosZac20, 3.c], we see that the condition $\mathcal {L} \subset \mathcal {L}^{\vee }$ amounts to

(5.1.3)\begin{equation} X^{t} S_1 X - 2 \pi S X=0,\quad X^{t} S_2 X + 2 SX =0. \end{equation}

Similarly, the condition ${\mathcal {L}}^\vee \subset u^{-1}{\mathcal {L}}$ amounts to

(5.1.4)\begin{equation} Y^{t} S_1 Y + 2 (S_2 Y + \pi S_1 Y)=0,\quad Y^{t} S_2 Y - 2\pi (S_2 Y + \pi S_1 Y ) =0. \end{equation}

In the above, $S_1$, $S_2$, are the matrices with $S=S_1+\pi S_2=(\langle e_i, e_j\rangle )_{i, j}$ as in § 2.2.

(d) Finally, $\bar {\mathcal {L}}$ and $\bar {\mathcal {L}}^\vee$ should be $u$-stable. This translates to

\[ X^2=0,\quad Y^2=0. \]

However, these are implied by equations (5.1.1) above.

Denote by $J_m$ the unit antidiagonal matrix of size $m$,

We now set $X=(x_{ij})$, $Y=(y_{ij})$, and denote by $\mathfrak I^{\text {naive}}$ the ideal of the polynomial ring $\breve {\mathcal {O}}[(x_{ij}), (y_{ij})]$ in $2d^2$ variables, which is generated by the entries of the above relations (5.1.1)–(5.1.4). Also denote by $U^{\text {naive}}$ the corresponding scheme over ${\rm Spec}\,(\breve {\mathcal {O}})$ given by pairs of $d \times d$ matrices $X$, $Y$ which satisfy these relations (for more details see [Reference ZachosZac20, § 3]). Set

\[ {U}^{\text{naive}}={\rm Spec}\,(\breve{\mathcal{O}}[X, Y]/\mathfrak I^{\rm naive}) \]

which is an open subscheme of ${\rm M}^{\rm naive}({\Lambda })$. Proposition 4.2.2 realizes ${\rm M}^{\rm loc}({\Lambda })$ as a closed subscheme of ${\rm M}^{\rm naive}({\Lambda })$, and we can set

\[ U=U^{\rm naive}\cap {\rm M}^{\rm loc}({\Lambda})={\rm Spec}\,(\breve{\mathcal{O}}[X, Y]/\mathfrak I), \]

with $\mathfrak I^{\rm naive}\subset \mathfrak I$. The scheme $U$ is an open affine subscheme of ${\rm M}^{\rm loc}({\Lambda })$. Denote by $\breve {\mathcal {O}}(U^{\rm naive})$ (respectively, $\breve {\mathcal {O}}(U)$) the affine coordinate rings of $U^{\rm naive}$ (respectively, $U$) over $\breve {\mathcal {O}}$.

Following [Reference ZachosZac20, 3.c], we distinguish two cases.

(I) The integers $d$ and $\delta$ have the same parity (cases (1) and (4) of § 2.2). We then break up the matrix $X$ into blocks as follows. We write

(5.1.5)\begin{equation} X =\left[\begin{array}{c|c|c} D_1 & C_1 & D_2 \\ \hline B_1 & A & B_2 \\ \hline D_3 & C_2 & D_4 \end{array}\right], \end{equation}

where $A$ is of size $\delta \times \delta$, and $D_1$, $D_2$, $D_3$, $D_4$ are of size $(n-r)\times (n-r)$. (Recall that $d=2n$, $\delta =2r$, or $d=2n+1$, $\delta =2r+1$.) We set

\[ {\mathrm {T}}(B_1\mid B_2)={\rm Tr}(B_2 J_{n-r} B_1^{t}J_{\delta}). \]

(II) The integers $d$ and $\delta$ have different parity (cases (2) and (3) of § 2.2). In this case, we again decompose the matrix $X$ into nine blocks as above, but the recipe for the dimensions of these blocks is somewhat different. In order to define the submatrices $A$, $B_{i}$, $C_j$, $D_j$, giving the block decomposition of $X$ we set

\[ r' =\begin{cases} r & \mbox{if } \delta =2r \\ r+1 & \mbox{if } \delta=2r+1. \end{cases} \]

Then we write the matrix $X$ as before, with blocks $A$ of size $(\delta +1) \times (\delta +1)$, and $D_1$, $D_2$, $D_3$, $D_4$ of size $(n-r') \times (n-r')$.

We denote by $A'$ the $\delta \times \delta$ matrix which is obtained from $A$ by erasing the part that is in the $(n+1)$th row and $(n+1)$th column of $X$. Similarly we denote by $B'_1,B'_2$ the $\delta \times (n- r')$ matrices which are obtained from $B_1$, $B_2$ by erasing the part that is on the $(n+1)$th row of $X$. Lastly, we denote by $E$ the $(r'+1 )$th column of $A$ and $E'$ the $(r'+1 )$th column of $A$ with the $(n+1)$th entry erased. (Recall that $d=2n$, $\delta =2r+1$, or $d=2n+1$, $\delta =2r$.) We set

\[ \mathrm T(B'_1\mid E'\mid B'_2)= {\rm Tr}( ( B'_2 J_{n-r'} (B'_1)^{t} +\tfrac{1}{2} E' (E')^{t})J_{\delta} ). \]

Finally, for simplicity, we set

\[ Z= \begin{cases} [B_1|B_2] & \hbox{if }d\equiv\delta\, {\rm mod}\ 2\\ [B_1'|E'|B'_2] & \hbox{if } d\not\equiv\delta\, {\rm mod}\ 2. \end{cases} \]

Then $Z=(z_{ij})\in {\rm Mat}_{\delta \times (d-\delta )}$, in both cases.

Theorem 5.1.6 [Reference ZachosZac20]

The inclusion $\breve {\mathcal {O}}[Z]\to \breve {\mathcal {O}}[X, Y]$ induces isomorphisms

(5.1.7)\begin{align} {\breve{\mathcal{O}}[Z]}/{( \wedge^2 Z, {\mathrm {T}}(Z)+ 2 \pi)}&\xrightarrow{\sim} \breve{\mathcal{O}}(U) , \end{align}

(5.1.8)\begin{align} {\breve{\mathcal{O}}[Z]}/{( \wedge^2 Z, ({\mathrm {T}}(Z) + 2 \pi)\cdot Z)} &\xrightarrow{\sim} \breve{\mathcal{O}}(U^{\rm naive}) . \end{align}

5.2

This is essentially contained in [Reference ZachosZac20] but, for completeness, we will also give the argument below. Before we do that, we discuss some corollaries.

(A) By an explicit calculation, we find that

\[ \mathrm T(Z)=\frac{1}{2}\sum_{1\leq i\leq \delta, 1\leq j\leq d-\delta} z_{i\ d-\delta+1-j}\, z_{\delta+1-i\ j}. \]

(The same expression is valid in both cases, of same (I), or different (II), parity.) The result stated in the introduction follows.

Denote by $\mathscr D^{2}_{\delta \times (d-\delta )}=\{Z\mid \wedge ^2Z=0\}\subset {\rm Mat}_{\delta \times (d-\delta )}$ the ‘determinantal’ subscheme of the affine space of matrices $Z$ over ${\rm Spec}\,(\breve {\mathcal {O}})$.

Theorem 5.2.1 The affine chart $U\subset {\rm M}^{\rm loc}({\Lambda })$ is isomorphic to the closed subscheme $\mathscr D_{\rm T}$ of the determinantal scheme $\mathscr D^{2}_{\delta \times (d-\delta )}$ which is defined by the quadratic equation

\[ \sum_{1\leq i\leq \delta,\ 1\leq j\leq d-\delta} z_{i\ d-\delta+1-j}\, z_{\delta+1-i\ j}=-4\pi. \]

(B) By Theorem 5.1.6,

(5.2.3)\begin{equation} \breve{\mathcal{O}}(U)=\breve{\mathcal{O}}(U^{\rm naive})/({\rm T}(Z)+2\pi), \end{equation}

where we slightly abuse notation by denoting by ${\rm T}(Z)+2\pi$ the image of this element in the quotient ring $\breve {\mathcal {O}}(U^{\rm naive})$. We can easily check that the annihilator of the element ${\rm T}(Z)+2\pi$ in the coordinate ring $\breve {\mathcal {O}}(U^{\rm naive})$ is the ideal $(Z)$ generated by the entries of $Z$. We obtain an exact sequence

\[ 0\to \breve{\mathcal{O}}(U^{\rm naive})\to \breve{\mathcal{O}}(U)\times \breve{\mathcal{O}}\xrightarrow{\ } k\to 0, \]

where the second map is the difference of the reductions modulo the maximal ideals $(Z)+ (\pi )$ and $(\pi )$. Hence,

\[ \breve{\mathcal{O}}(U^{\rm naive})=\breve{\mathcal{O}}(U)\times_{k} \breve{\mathcal{O}} \]

where, on the right-hand side, we have the fibered product of rings. This exhibits $U^{\rm naive}$ as the push-out

It easily follows that ${\rm M}^{\rm naive}({\Lambda })$ is also a push-out:

(5.2.4)

We now give the proof of Theorem 5.1.6.

Proof of Theorem 5.1.6 This is a variation of the work in [Reference ZachosZac20]. We explain a simpler version of the argument. (We will also omit some of the explicit calculations.)

We start with the case (I) of same parity $d\equiv \delta \, {\rm mod}\, 2$.

First, we prove that $\breve {\mathcal {O}}(U^{\rm naive})$ (respectively, $\breve {\mathcal {O}}(U)$) are quotient rings of $\breve {\mathcal {O}}[B_1, B_2]$.

Let $\mathfrak R$ be the ideal generated by (the entries of) the following elements:

1. (i) $D_1 +\frac {1}{2} J_{n-r} B_2^{t}J_{\delta } B_1,$

2. (ii) $D_2 +\frac {1}{2} J_{n-r} B_2^{t}J_{\delta } B_2 ,$

3. (iii) $D_3 +\frac {1}{2} J_{n-r} B_1^{t}J_{\delta } B_1 ,$

4. (iv) $D_4 +\frac {1}{2} J_{n-r} B_1^{t}J_{\delta } B_2 ,$

5. (v) $C_1 +\frac {1}{2} J_{n-r} B_2^{t}J_{\delta } A ,$

6. (vi) $C_2 +\frac {1}{2} J_{n-r} B_1^{t}J_{\delta } A$,

7. (vii) $A - B_2 J_{n-r} B_1^{t}J_{\delta }$,

8. (viii) $Y+X^t$.

We observe that

\[ \mathfrak R \subset \mathfrak I^{\rm naive}. \]

Indeed, relations (i)–(vi) are implied from the relation $X^{t} S_1 X - 2 \pi S X\in \mathfrak I^{\rm naive}$ (5.1.3), relation (vii) from (5.1.1) and (5.1.4), and (viii) from (5.1.1).

Hence, relations (i)–(viii) express each block $A$, $D_i$, $C_j$, of $X$ and $Y$, in terms of $B_1$ and $B_2$, modulo $\mathfrak I^{\rm naive}$. It follows that $\breve {\mathcal {O}}(U^{\rm naive})$, and therefore also $\breve {\mathcal {O}}(U)$, is a quotient of $\breve {\mathcal {O}}[B_1, B_2]$. In fact, since $\mathfrak I^{\rm naive}$ contains all the $2\times 2$ minors of $X$, it follows that $\breve {\mathcal {O}}(U^{\rm naive})$ and $\breve {\mathcal {O}}(U)$ are also quotients of $\breve {\mathcal {O}}[B_1, B_2]/(\wedge ^2[B_1\mid B_2])$. This last ring is the affine coordinate ring of the cone over the Segre embedding of ${\mathbb {P}}^{\delta -1}\times {\mathbb {P}}^{d-\delta -1}$ over $\breve {\mathcal {O}}$ and so it is integral and flat of relative dimension $d-1$ over $\breve {\mathcal {O}}$.

We now continue to uncover additional relations in $\mathfrak I^{\rm naive}$. As is observed in [Reference ZachosZac20, 4.a], the condition $\wedge ^2 X=0$ easily gives

(5.2.5)\begin{equation} AB_1={\rm Tr}(A)B_1,\quad AB_2={\rm Tr}(A)B_2. \end{equation}

By [Reference ZachosZac20, Lemma 4.4], the relation $\wedge ^2 X=0$ implies that $B_1^t J_{n-r} B_2$ is symmetric, so, modulo $\mathfrak I^{\rm naive}$, we have

\[ B_1^t J_{n-r} B_2=B_2^t J_{n-r} B_1. \]

By looking at the appropriate blocks, we see that relations (5.1.3) imply

\[ A^t J_\delta B_1+2\pi J_\delta B_1\in \mathfrak I^{\rm naive},\quad A^t J_\delta B_2+2\pi J_\delta B_2 \in \mathfrak I^{\rm naive}. \]

Now $A - B_2 J_{n-r} B_1^{t}J_{\delta } \in \mathfrak I^{\rm naive}$ (relation (vii) above), so, modulo $\mathfrak I^{\rm naive}$,

\begin{align*} A^t J_\delta B_1+2\pi J_\delta B_1&= (B_2 J_{n-r} B_1^{t}J_{\delta})^t J_\delta B_1+2\pi J_\delta B_1\\ &= J_\delta B_1 J_{n-r} B_2^t J_\delta B_1+2\pi J_\delta B_1\\ &= J_\delta A B_1+2\pi J_\delta B_1\\ &= J_\delta({\rm Tr}(A)+2\pi)B_1, \end{align*}

where the last identity uses (5.2.5). This implies that $({\rm Tr}(A)+2\pi )B_1\in \mathfrak I^{\rm naive}.$ Similarly, starting from

\[ A^t J_\delta B_2+2\pi J_\delta B_2\in \mathfrak I^{\rm naive}, \]

we obtain $({\rm Tr}(A)+2\pi )B_2\in \mathfrak I^{\rm naive}.$ Since ${\rm T}(B_1\,|\,B_2)={\rm Tr}(A)$ modulo $\mathfrak I^{\rm naive}$, this gives that $\breve {\mathcal {O}}(U^{\rm naive})$ is a quotient of $\breve {\mathcal {O}}[B_1, B_2]/(\wedge ^2 [B_1\mid B_2], ({\rm T}(B_1\mid B_2)+2\pi )[B_1\mid B_2])$.

Observe that $B_1=B_2=0$ gives $X=Y=0$, which corresponds to ${\mathcal {L}}={\mathbb {L}}$. This point does not belong to the generic fiber of the $\breve {\mathcal {O}}$-flat ${\rm M}^{\rm loc}$. Hence, we find that $\breve {\mathcal {O}}(U)$ is a quotient of ${\mathcal {A}}=\breve {\mathcal {O}}[B_1, B_2]/(\wedge ^2 [B_1\mid B_2], {\rm T}(B_1\mid B_2)+2\pi )$. The ring ${\mathcal {A}}$ is the coordinate ring of a hypersurface in the integral Cohen–Macaulay $\breve {\mathcal {O}}[B_1, B_2]/(\wedge ^2 [B_1\mid B_2])$; we can easily see as in [Reference ZachosZac20, § 5] that ${\mathcal {A}}$ is integral of dimension $d-2$ and $\breve {\mathcal {O}}$-flat. Since $U$ also shares both these properties and $\breve {\mathcal {O}}(U)$ is a quotient of ${\mathcal {A}}$, it follows that $\breve {\mathcal {O}}(U)={\mathcal {A}}$, as we wanted. The result for $\breve {\mathcal {O}}(U^{\rm naive})$ also quickly follows. Indeed, $\breve {\mathcal {O}}(U^{\rm naive})$ is a quotient of

\[ {\mathcal{A}}_+:=\breve{\mathcal{O}}[B_1, B_2]/(\wedge^2 [B_1\mid B_2], ({\rm T}(B_1\mid B_2)+2\pi)[B_1\mid B_2]). \]

However, this quotient has to be large enough to also allow both algebras $\breve {\mathcal {O}}(U)={\mathcal {A}}$ and $\breve {\mathcal {O}}={\mathcal {O}}[B_1, B_2]/([B_1\mid B_2])$ to appear as quotients. The corresponding spectrum has to support a morphism from the push-out of $U$ and ${\rm Spec}\,(\breve {\mathcal {O}})$, glued at the point $X=Y=\pi =0$. But, as in § 5.2 above, this push-out is the spectrum of ${\mathcal {A}}_+$, so

\[ \breve{\mathcal{O}}(U^{\rm naive})={\mathcal{A}}_+=\breve{\mathcal{O}}[B_1, B_2]/(\wedge^2 [B_1\mid B_2], ({\rm T}(B_1\mid B_2)+2\pi)[B_1\mid B_2]), \]

as we wanted.

The case (II) of different parity is similar. The role of $A$ is now played by $A'$ and relation (vii) above is replaced by

\[ A'-(B'_2 J_{n-r'} (B'_1)^{t} + \tfrac{1}{2} E'(E')^{t}) J_\delta \in \mathfrak I^{\rm naive} \]

which gives ${\rm Tr}(A')={\rm T}(B'_1\mid E'\mid B_2')$ modulo $\mathfrak I^{\rm naive}$. See also the proof of [Reference ZachosZac20, Theorem 3.3] for more details.

5.3 The blow-up of the local model ${\rm M}^{\rm loc}$

Let

\[ r^{\rm bl}: {\rm M}^{\rm bl}({\Lambda})\to {\rm M}^{\rm loc}({\Lambda}) \]

be the blow-up of ${\rm M}^{\rm loc}({\Lambda })$ at the closed point $*$ of its special fiber that corresponds to ${\mathcal {L}}={\mathbb {L}}$. We will now show Theorem 1.3.1.

Proof. By Theorem 5.2.1, it is enough to show the conclusion of the theorem for the blow-up $\widetilde {\mathscr D}_{\rm T}$ of ${\mathscr D}_{\rm T}$ at the (maximal) ideal given by $(z_{ij})$. For simplicity, we write $\mathscr D$ for the determinantal scheme ${\mathscr D}^2_{\delta \times (d-\delta )}$ over ${\rm Spec}\,(\breve {\mathcal {O}})$. This is the affine cone over the Segre embedding

\[ ({\mathbb{P}}^{\delta-1}\times {\mathbb{P}}^{d-\delta-1})_{\breve{\mathcal{O}}}\hookrightarrow {\mathbb{P}}^{\delta(d-\delta)-1}_{\breve{\mathcal{O}}}. \]

Also, we set

\[ {\rm T}=\frac{1}{2}\sum_{1\leq i\leq \delta,\ 1\leq j\leq d-\delta} z_{i\ d-\delta+1-j}\, z_{\delta+1-i\ j}. \]

Let us consider the blow-up

\[ \widetilde {\mathscr D} \longrightarrow {\mathscr D} \]

of the determinantal scheme over ${\rm Spec}\,(\breve {\mathcal {O}})$ along the vertex of the cone, that is, along the subscheme defined by the ideal $(z_{ij})$. Then the blow-up $\widetilde {\mathscr D}_{\rm T}$ is isomorphic to the strict transform of the hypersurface $\mathscr D_{\rm T}\subset \mathscr D$ given by ${\rm T}+2\pi =0$. By definition, $\widetilde {\mathscr D}$ is a closed subscheme of ${\mathscr D}\times _{\breve {\mathcal {O}}} {\mathbb {P}}^{\delta (d-\delta )-1}_{\breve {\mathcal {O}}}$, and in fact

\[ \widetilde {\mathscr D}= \{(z_{ij}, u_{i,j})\mid (z_{ij}u_{i',j'}-z_{i'j'}u_{i,j}), \wedge^2(z_{ij})=0,\ \wedge^2(u_{i,j})=0\} \subset {\mathscr D}\times_{\breve{\mathcal{O}}} {\mathbb{P}}^{\delta(d-\delta)-1}_{\breve{\mathcal{O}}} \]

where $(u_{i,j})$ are homogeneous coordinates on ${\mathbb {P}}^{\delta (d-\delta )-1}_{\breve {\mathcal {O}}}$. Let $V_{s, t}$ be the open affine of $\widetilde {\mathscr D}$ over which the image of $z_{s t}$ generates the pull-back of the ideal $(z_{ij})$. Then

\[ V_{s, t}\simeq {\rm Spec}\,(\breve{\mathcal{O}}[(u_{i,j})_{1\leq i\leq \delta, 1\leq j\leq d-\delta}]/((u_{i,j}-u_{s, j} u_{i, t})_{i, j}, u_{s, t}-1). \]

The intersection $V_{s, t}\cap \widetilde {\mathscr D}_{\rm T}$ is obtained by substituting $z_{ij}=u_{i,j}z_{st}$ and $u_{i,j}=u_{s, j}u_{i,t}$, for all $i$, $j$, in the equation ${\rm T}=-2\pi$. This amounts to setting

\[ z_{ij}=u_{s,j}u_{i,t}z_{st} \]

and gives

\[ 4\pi+z_{st}^2 \bigg(\sum_{1\leq i\leq \delta,\ 1\leq j\leq d-\delta } u_{s, d-\delta+1-j} u_{i, t} u_{s,j} u_{\delta+1-i, t}\bigg)=0. \]

This is

(5.3.1)\begin{equation} 4\pi+z_{st}^2 \bigg(\sum_{i=1}^\delta u_{i,t}u_{\delta+1-i, t}\bigg) \bigg(\sum_{j=1}^{d-\delta} u_{s,j} u_{s, d-\delta+1-j}\bigg)=0. \end{equation}

Note that, since $u_{s,t}=1$, the two sums in the line above are

\[ S_1=u_{\delta+1-s, t} +\sum_{i\neq s} u_{i,t}u_{\delta+1-i, t},\quad S_2=u_{s, d-\delta+1-t} +\sum_{j\neq t} u_{s,j} u_{s, d-\delta+1-j}. \]

(If $\delta _*=1$, that is, $\delta =1$ or $d-\delta =1$, then one of the sums is equal to $u_{s, t}=1$.) If $\delta _*\geq 2$, we see that $u\mapsto z_{st}$, $x\mapsto -S_1/2$, $y\mapsto S_2/2$ defines a smooth morphism

\[ V_{s, t}\cap \widetilde {\mathscr D}_{\rm T}\longrightarrow {\rm Spec}\,(\breve{\mathcal{O}}[u, x, y]/(u^2xy-\pi)). \]

If $\delta _*=1$, we similarly obtain a smooth morphism to ${\rm Spec}\,(\breve {\mathcal {O}}[u, x]/(u^2x-\pi ))$.

6.1 A resolution via additional lines

We first define a scheme ${\mathcal {M}}^{\rm naive}={\mathcal {M}}^{\rm naive}({\Lambda })$ over ${\rm Spec}\,({\mathcal {O}})$ with ${\mathcal {G}}$-action and a ${\mathcal {G}}$-equivariant morphism $r: {\mathcal {M}}^{\rm naive}({\Lambda })\longrightarrow {\rm M}^{\rm naive}({\Lambda })$.