\label{cor:Grasssmannian}
Fix integers $1 \leq \ell < n$ and let $X = \mathrm{Gr}_{\ell,n}(\mathbb{R})$ be the Grassmann variety of $\ell$-dimensional subspaces in $\mathbb{R}^n$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{cor:eq_Grassmannian}
\mathcal{N}_{c, \beta}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}

\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \label{thm:Effective}
Suppose the action of $G$ on $\Omega = G/\G$ has a spectral gap. Let $A = \{a(y) : y \in \R_+^{\times} \}$ be an $\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \geq 1$ such that for every compact subset $Q \subset \Omega$, for all $f \in C^{\infty}(K)$, $\phi \in C_c^\infty(\Omega)$, $x \in Q$, and $y \geq 1$, we have
\begin{equation} \label{eq:Single-Eq}
\int_K f(k) \phi \bigl(a(y) k x \bigr) \dd \mu_K(k) = \int_K f \dd \mu_K \int_{\Omega} \phi \dd \mu_{\Omega} + O \big ( y^{-c} \cS_r(f) \cS_r(\phi) \big ),
\end{equation}
and, for all $f \in C^{\infty}(K)$, $\phi_1, \phi_2 \in C_c^\infty(\Omega)$, $x_1, x_2 \in Q$,
and $y_2 \geq y_1 \geq 1$, we have
\begin{multline} \label{eq:Double-Eq}
\int_K f(k) \phi_{1}\bigl(a(y_1) k x_1 \bigr) \phi_{2} \bigl(a(y_2) k x_2 \bigr) d \mu_K(k) =  \int_K f \dd \mu_K \int_{\Omega} \phi_1 \dd \mu_{\Omega}  \int_{\Omega} \phi_2 \dd \mu_{\Omega} \\ + O \big ( \min \{y_1, y_2/y_1 \}^{-c} \cS_r(f) \cS_r(\phi_1) \cS_r(\phi_2) \big ),
\end{multline}
where the implicit constant in $O(\cdot)$ depends only on $Q$.
\end{thmx}

\begin{equation} \label{eq:SchmidtCountingFunction}
\cN_\psi(x,T) = \# \left \{ (p,q) \in \Z \times \N : 0 \leq q x - p < \psi(q), \, 1 \leq q < T \right \}
\end{equation}

\begin{equation} \label{eq:primitive_Siegel_Transform}
\forall \, g \in \mathrm{SL}_n(\mathbb{R}), \qquad S f(g\Z^n) = \sum_{\bm{v} \in \cP(\Z^n)} f (g \bm{v}).
\end{equation}

\begin{thmx} [$L^2$-integrability] \label{thm:L2}
Suppose that the Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{2}(\Omega)$. Then
\[
\forall \, w \in W, \quad X^*(\bL_w^\circ)_{\Q} = \{1\}.
\]
In particular, the parabolic $\Q$-subgroup $\bP$ is maximal and the simple root $\alpha \in \Delta$, satisfying that $\bP = \bP_{\Delta \smallsetminus \{\alpha \}}$, has at most one neighbor in the associated Dynkin diagram.
\end{thmx}

\begin{equation} \label{eq:Norm}
\cS_r(\phi) = \sum_{\deg(\cD) \leq r} \left \|  \cD \phi \right \|_{\infty},
\end{equation}

\begin{equation} \label{eq:primitive_Siegel_Transform_variance}
\int_{\Omega} \left| S f - \frac{1}{\zeta(n)} \int_{\R^n} f \, \dd \lambda_{\R^n} \right|^2 \dd \mu_{\Omega} \, \ll \, \int_{\R^n} |f|^2 \, \dd \lambda_{\R^n}.
\end{equation}

The Siegel transform, introduced in 1945 by Siegel \cite{Siegel45}, maps a function of sufficient decay on the Euclidean space $\mathbb{R}^n$ to a function on the moduli space of unimodular lattices $\Omega = \mathrm{SL}_n(\mathbb{R}) / \mathrm{SL}_n(\mathbb{Z})$. Let ${\mathcal P}(\mathbb{Z}^n)$ denote the set of primitive elements of $\mathbb{Z}^n$ and let $B_c^{\infty}(\mathbb{R}^n)$ be the space of Borel measurable bounded compactly supported functions $f : \mathbb{R}^n \rightarrow \mathbb{C}$. Then, for every $f \in B_c^{\infty}(\mathbb{R}^n)$, the primitive Siegel transform $S f : \Omega \rightarrow \mathbb{C}$ of $f$ is defined by
\begin{equation} \label{eq:primitive_Siegel_Transform}
\forall \, g \in \mathrm{SL}_n(\mathbb{R}), \qquad S f(g\mathbb{Z}^n) = \sum_{\bm{v} \in {\mathcal P}(\mathbb{Z}^n)} f (g \bm{v}).
\end{equation}
Let $\mu_{\Omega}$ be the unique $\SL_{n}(\mathbb{R})$-invariant probability measure on $\Omega$, let $\zeta$ be the Riemann zeta function and let $\lambda_{\mathbb{R}^n}$ be the usual Lebesgue measure on $\mathbb{R}^n$. Siegel's mean value formula \cite{Siegel45} expresses the average of $Sf$ in terms of the average of $f$:
\begin{equation} \label{eq:primitive_Siegel_Transform_formula}
\int_{\Omega} S f \, \,\mathrm{d} \mu_{\Omega} = \frac{1}{\zeta(n)} \int_{\mathbb{R}^n} f \, \,\mathrm{d} \lambda_{\mathbb{R}^n}.
\end{equation}
Later, extending Siegel’s result, Rogers \cite{Rogers55} proved a $k$-th moment formula for the Siegel transform for $k$ up to $n-1$. A remarkable application of the second moment formula to the geometry of numbers was given by Schmidt \cite{Schmidt60b}, who derived an asymptotic formula for counting lattice points in an expanding family of sets in $\mathbb{R}^n$ from the variance bound
\begin{equation} \label{eq:primitive_Siegel_Transform_variance}
\int_{\Omega} \left| S f - \frac{1}{\zeta(n)} \int_{\mathbb{R}^n} f \, \,\mathrm{d} \lambda_{\mathbb{R}^n} \right|^2 \,\mathrm{d} \mu_{\Omega} \, \ll \, \int_{\mathbb{R}^n} |f|^2 \, \,\mathrm{d} \lambda_{\mathbb{R}^n}.
\end{equation}
In other words, the \emph{centered} Siegel transform $\overline{S} f = Sf - 1/\zeta(n) \int_{\mathbb{R}^n} f \, \,\mathrm{d} \lambda_{\mathbb{R}^n}$ extends to a bounded linear operator
\[
\overline{S} : L^2(\mathbb{R}^n) \rightarrow L^2(\Omega).
\]
The variance bound \eqref{eq:primitive_Siegel_Transform_variance} also yields an alternative proof of Schmidt’s strengthening \cite{Schmidt60a} of Khintchine’s theorem \cite{Khintchine26} in metric Diophantine approximation on $\mathbb{R}^n$ and its projective counterpart $\mathbb{P}(\mathbb{R}^n)$. There has been an active line of research extending classical results in Diophantine approximation from Euclidean space to other varieties, such as spheres \cite{AG22, KM15, KY23, Ouaggag23}, projective quadrics \cite{SK18, deSaxce22b, FKMS22}, Grassmannians \cite{deSaxce22a}, and more general flag varieties \cite{deSaxce20}.

The purpose of this paper is to study fundamental integrability properties of a natural extension of the Siegel transform \eqref{eq:primitive_Siegel_Transform} from the Euclidean space to the setting of generalized flag varieties. Our results have an application to metric Diophantine approximation on rank-one flag varieties. The proof of this application relies in addition on the effective single and double equidistribution property for expanding orbits of maximal compact subgroups, a result of independent interest.

\subsection{Main results}
Let ${\bf G}$ be a connected simply-connected almost $\mathbb{Q}$-simple $\mathbb{Q}$-group and let ${\bf P}$ be a proper parabolic $\mathbb{Q}$-subgroup of ${\bf G}$. We denote algebraic varieties defined over $\mathbb{Q}$ by bold letters and their sets of real points by ordinary letters. For instance, we write $G = {\bf G}(\mathbb{R})$ to denote the group of real points of ${\bf G}$. Let $\Gamma \subset {\bf G}(\mathbb{Q})$ be an arithmetic subgroup of $G$. Let $\pi_{\chi} : {\bf G} \rightarrow \GL(\bV_{\chi})$ be an irreducible representation defined over $\mathbb{Q}$ which is generated by a line $\bD_{\chi}$ defined over $\mathbb{Q}$ of highest weight $\chi$ such that ${\bf P} = \mathrm{Stab}_{{\bf G}} (\bD_{\chi})$ (Section~\ref{sec:Reps}). In particular, the space of real points $X = {\bf X}(\mathbb{R})$ of the generalized flag variety ${\bf X} = {\bf G} / {\bf P}$ embeds into the projective space $\mathbb{P}(V_{\chi})$. We fix a highest weight vector $\bm{e}_{\chi} \in \bD_{\chi}(\mathbb{Q})$ and define $\widetilde{X}$ to be the orbit $\widetilde{X} = G \, \bm{e}_{\chi} \subset V_{\chi}$. We refer to $\widetilde{X}$ as the \emph{cone over $X$ relative to $\chi$}. Fix a $\Gamma$-stable lattice $\bV_{\chi}(\mathbb{Z}) \subset \bV_{\chi}(\mathbb{Q})$ of $V_{\chi}$ and denote by $\cP_{\chi}$ the set of primitive elements of $\bV_{\chi}(\mathbb{Z}) \cap \widetilde{X}$. Let $B_c^{\infty}(\widetilde{X})$ be the space of Borel measurable bounded compactly supported complex-valued functions $f : \widetilde{X} \rightarrow \mathbb{C}$.
\begin{definition} [Siegel transform]
For every $f \in B_c^{\infty}(\widetilde{X})$, we define the \emph{Siegel transform} $S_{\chi} f : \Omega \rightarrow \mathbb{C}$ of $f$ by
\begin{equation*} \label{def:Siegel-Transform}
\forall \, g \in G, \quad S_{\chi} f(g \Gamma) = \sum_{\bm{v} \in \cP_{\chi}} f(g \bm{v}).
\end{equation*}
\end{definition}

Let $\mu_{\Omega}$ be the unique $G$-invariant Borel probability measure on the homogeneous space $\Omega = G/\Gamma$. We will answer the question: \emph{For any $p = 1,2, \infty$, what are necessary and sufficient conditions for $S_{\chi}$ to map $B_c^{\infty}(\widetilde{X})$ into $L^p(\Omega)$?}

In our first result, the equivalences $(1)$ - $(4)$ are likely known to experts; the formula \eqref{eq:WeilIntegrationFormula*} below is a consequence of a general integration formula due to Weil \cite[Theorem~2.51]{Folland15}. Let ${\bf L} = \mathrm{Stab}_{{\bf G}} (\bm{e}_{\chi}) \subset {\bf P}$.

\begin{thmx} [$L^1$-integrability] \label{thm:L1}
The following assertions are equivalent.
\begin{enumerate}[label=(\arabic*)]
\item The Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^1(\Omega)$.
\item There exists a unique (up to scaling) $G$-invariant Radon measure $\lambda_{\widetilde{X}}$ on $\widetilde{X}$, the Siegel transform $S_\chi$ extends to a bounded operator $S_{\chi} : L^1(\widetilde{X}) \rightarrow L^1(\Omega)$, and $\lambda_{\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:
\begin{equation} \label{eq:WeilIntegrationFormula*}
\forall \, f \in L^1(\widetilde{X}), \quad \int_{\Omega} S_\chi f \, \,\mathrm{d} \mu_{\Omega} = \int_{\widetilde{X}} f \, \,\mathrm{d} \lambda_{\widetilde{X}}.
\end{equation}
\item The Lie group $L = {\bf L}(\mathbb{R})$ is unimodular and $\G_L = \Gamma \cap L$ is a lattice in $L$.
\item The parabolic $\mathbb{Q}$-subgroup ${\bf P}$ of ${\bf G}$ is maximal.
\item There exists $\varepsilon > 0$ such that $S_\chi$ maps $B_{c}^{\infty}(\widetilde{X})$ into $L^{1+\varepsilon}(\Omega)$.
\end{enumerate}
\end{thmx}

\begin{thmx} [$L^{\infty}$-integrability] \label{thm:Linfty}
The following assertions are equivalent.
\begin{enumerate} [label=(\arabic*)]
\item The Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{\infty}(\Omega)$.
\item The $\mathbb{Q}$-rank of $\mathbf{G}$ is $1$.
\item The discrete group $\G_L$ is a cocompact lattice in $L$.
\end{enumerate}
\end{thmx}

Let $\bP_0$ be a minimal parabolic $\mathbb{Q}$-subgroup of ${\bf G}$ contained in ${\bf P}$ and let ${\bf T}$ be a maximal $\mathbb{Q}$-split torus of ${\bf G}$ contained in $\bP_0$. Let $\Phi$ be the root system of ${\bf G}$ relative to ${\bf T}$ and let $\Delta \subset \Phi$ be the corresponding set of simple roots. For each subset $\theta$ of $\Delta$, write $\bP_{\theta}$ for the associated standard parabolic $\mathbb{Q}$-subgroup of ${\bf G}$. Let $W$ be the Weyl group of ${\bf G}$ relative to ${\bf T}$. For every $w \in W$, we define $\bL_w = {\bf L} \cap x_w {\bf L} x_w^{-1}$, where $x_w \in \cN_{{\bf G}}({\bf T})(\mathbb{Q})$ is a representative of $w$, and denote by $X^*(\bL_w^{\circ})_{\mathbb{Q}}$ the group of $\mathbb{Q}$-characters of its identity component.

\begin{thmx} [$L^2$-integrability] \label{thm:L2}
Suppose that the Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{2}(\Omega)$. Then
\[
\forall \, w \in W, \quad X^*(\bL_w^\circ)_{\mathbb{Q}} = \{1\}.
\]
In particular, the parabolic $\mathbb{Q}$-subgroup ${\bf P}$ is maximal and the simple root $\alpha \in \Delta$, satisfying that ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha \}}$, has at most one neighbor in the associated Dynkin diagram.
\end{thmx}

We were unable to determine whether the converse statement is true: \emph{Assuming that for every $w \in W$, we have $X^*(\bL_w^\circ)_{\mathbb{Q}} = \{1\}$, does the Siegel transform $S_\chi$ map $B_c^{\infty}(\widetilde{X})$ into $L^2(\Omega)$?}

For every $f \in B_c^{\infty}(\widetilde{X})$, define the \emph{centered Siegel transform} $\overline{S}_{\chi} f : \Omega \rightarrow \mathbb{C}$ of $f$ by
\[
\forall \, g \in G, \qquad \overline{S}_{\chi} f(g\Gamma) = S_{\chi}f(g\Gamma) - \int_{\widetilde{X}} f \, \,\mathrm{d} \lambda_{\widetilde{X}}.
\]
Beyond the case $p=q=1$ in Theorem~\ref{thm:L1}, it is natural to ask for which pairs $p,q\in[1,+\infty]$ the Siegel transform, or its centered counterpart, extends to a bounded linear operator $L^p(\widetilde{X}) \to L^q(\Omega)$. More specifically, we would like to include the following question, due to Saxc\'e, suggesting a fractional version of the variance bound \eqref{eq:primitive_Siegel_Transform_variance} that also takes into account point~(5) of Theorem~\ref{thm:L1} as well as Theorem~\ref{thm:L2}:
\emph{Assuming that the parabolic $\mathbb{Q}$-subgroup ${\bf P}$ is maximal, does there exist $\varepsilon>0$ such that the centered Siegel transform $\overline{S}_{\chi}$ extends to a bounded linear operator}
\[
\overline{S}_{\chi} : L^{1+\varepsilon}(\widetilde{X}) \to L^{1+\varepsilon}(\Omega)\,?
\]

\subsection{Effective equidistribution of maximal compact subgroup orbits}
The fact that the Siegel transform $S_{\chi}$ maps $B_c^{\infty}(\widetilde{X})$ into $L^{1+\varepsilon}(\Omega)$ for some small $\varepsilon > 0$ when the parabolic subgroup ${\bf P}$ is maximal (Theorem~\ref{thm:L1}), together with the effective single and double equidistribution property for translated orbits of maximal compact subgroups (Theorem~\ref{thm:Effective}), are the key analytic inputs for our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties. Before describing this application, let us state here our equidistribution result, which will be derived from an effective multiple equidistribution result for expanding translates of horospherical orbits due to Shi (see \cite[Theorem~1.5]{Shi21}). Let $K \subset G$ be a maximal compact subgroup, equipped with the Haar probability measure $\mu_K$. We write $\cS_r$ ($r \in \mathbb{N}^*)$ for the degree $r$ Sobolev norms on $C_c^{\infty}(\Omega)$ and $C^{\infty}(K)$ as defined in \eqref{eq:Norm}.

\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \label{thm:Effective}
Suppose the action of $G$ on $\Omega = G/\Gamma$ has a spectral gap. Let $A = \{a(y) : y \in \R_+^{\times} \}$ be an $\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \geq 1$ such that for every compact subset $Q \subset \Omega$, for all $f \in C^{\infty}(K)$, $\phi \in C_c^\infty(\Omega)$, $x \in Q$, and $y \geq 1$, we have
\begin{equation} \label{eq:Single-Eq}
\int_K f(k) \phi \bigl(a(y) k x \bigr) \,\mathrm{d} \mu_K(k) = \int_K f \,\mathrm{d} \mu_K \int_{\Omega} \phi \,\mathrm{d} \mu_{\Omega} + O \big ( y^{-c} \cS_r(f) \cS_r(\phi) \big ),
\end{equation}
and, for all $f \in C^{\infty}(K)$, $\phi_1, \phi_2 \in C_c^\infty(\Omega)$, $x_1, x_2 \in Q$,
and $y_2 \geq y_1 \geq 1$, we have
\begin{multline} \label{eq:Double-Eq}
\int_K f(k) \phi_{1}\bigl(a(y_1) k x_1 \bigr) \phi_{2} \bigl(a(y_2) k x_2 \bigr) d \mu_K(k) =  \int_K f \,\mathrm{d} \mu_K \int_{\Omega} \phi_1 \,\mathrm{d} \mu_{\Omega}  \int_{\Omega} \phi_2 \,\mathrm{d} \mu_{\Omega} \\ + O \big ( \min \{y_1, y_2/y_1 \}^{-c} \cS_r(f) \cS_r(\phi_1) \cS_r(\phi_2) \big ),
\end{multline}
where the implicit constant in $O(\cdot)$ depends only on $Q$.
\end{thmx}

\subsection{Application to Diophantine approximation on flag varieties}
Let us now state our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties, which uses Theorems~\ref{thm:L1} and \ref{thm:Effective} as inputs. Many classical results in Diophantine approximation on the real line $\mathbb{R}$ or in Euclidean space $\mathbb{R}^n$ admit a dynamical reinterpretation in terms of properties of certain diagonal orbits in the space of lattices $\Omega = \SL_n(\mathbb{R}) / \SL_n(\mathbb{Z})$; this is known as Dani's correspondence \cite{Dani85}. Via this dynamical reinterpretation and building on influential work of Margulis, Kleinbock and others \cite{KM96, KM98, KM99, FKMS22}, Saxc\'e \cite{deSaxce20} extended analogues of classical results to generalized flag varieties ${\bf X} = {\bf G} / {\bf P}$ defined over $\mathbb{Q}$. First examples of such varieties include projective $n$-space $\mathbb{P}^n(\mathbb{R})$, the Grassmann variety $\mathrm{Gr}_{\ell,n}(\mathbb{R})$ of $\ell$-dimensional subspaces of $\mathbb{R}^n$, projective quadric hypersurfaces (that is, the solution set in $\mathbb{P}^n(\mathbb{R})$ of a non-degenerate rational quadratic form in $n+1$ variables), and more general flag varieties, parametrizing flags of subspaces of a Euclidean space.

Let $\psi : \mathbb{N} \rightarrow (0,+\infty)$ be a non-increasing function. By Khintchine's theorem~\cite{Khintchine26}, the inequality
\[
0 \leq q x - p < \psi(q)
\]
has infinitely (resp. at most finitely) many solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ for almost every $x \in \mathbb{R}$, if the series $\sum_{q = 1}^{\infty} \psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \in \mathbb{R}$ and $T \geq 1$, he considered the counting function
\begin{equation} \label{eq:SchmidtCountingFunction}
\cN_\psi(x,T) = \# \left \{ (p,q) \in \mathbb{Z} \times \mathbb{N} : 0 \leq q x - p < \psi(q), \, 1 \leq q < T \right \}
\end{equation}
and showed that for almost every $x \in \mathbb{R}$, $\cN_\psi(x,T)$ is asymptotically equal to $\sum_{1\leq q < T} \psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\mathbb{R}^n$ of any dimension $n \geq 1$.

Our goal is to prove a version of this theorem, where the Euclidean space $\mathbb{R}^n$ is replaced by the space of real points $X = {\bf X}(\mathbb{R})$ of the generalized flag variety ${\bf X} = {\bf G} / {\bf P}$ defined over $\mathbb{Q}$. We assume that ${\bf P}$ is a maximal parabolic $\mathbb{Q}$-subgroup of ${\bf G}$ with abelian unipotent radical. In particular, ${\bf X}$ has $\mathbb{Q}$-rank $1$ and there exists a unique simple root $\alpha \in \Delta$ such that ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha\}}$. Let $Y$ be the unique element in the Lie algebra of ${\bf T}(\mathbb{R})$ such that
\[
\alpha(Y) = -1 \quad \text{and} \quad \beta(Y) = 0 \quad \text{for all } \beta \in \Delta \smallsetminus \{\alpha\}.
\]
We suppose that the element $\exp(Y)$ projects non-trivially to each simple factor of $G$. Let $K$ be a maximal compact subgroup of $G$. Let $\sigma_X$ be the unique $K$-invariant probability measure on $X$. We equip $X$ with a $K$-invariant Riemannian distance $d(\cdot, \cdot)$ and the set of rational points ${\bf X}(\mathbb{Q})$ with a height function $H_\chi$ associated to an irreducible rational representation $\pi_{\chi} : {\bf G} \rightarrow \GL(\bV_{\chi})$ which is generated by a unique rational line $\bD_{\chi}$ of highest weight $\chi$ such that $\mathrm{Stab}_{{\bf G}} (\bD_{\chi}) = {\bf P}$ (see Section \ref{sec:Reps}). By \cite[Th\'eor\`emes 2.4.5 et 3.2.1]{deSaxce20}, there exists a rational number $\beta_\chi \in \Q_{>0}$ such that, for every $c > 0$ and for $\sigma_X$-almost every $x \in X$, the inequality
\begin{equation} \label{eq:DiophantineExpo}
d(x,v) < c \, H_\chi (v)^{-\tau}
\end{equation}
admits infinitely (resp. at most finitely) many solutions $v \in {\bf X}(\mathbb{Q})$, if $\tau \leq \beta_\chi$ (resp. $\tau > \beta_\chi$). We refer to $\beta_\chi$ as the \emph{Diophantine exponent} of $X$ relative to $\chi$ and to $\tau \in [0,\beta_{\chi}]$ as an \emph{approximation exponent}.

In analogy to \eqref{eq:SchmidtCountingFunction}, for every constant $c > 0$, approximation exponent $\tau \in [0, \beta_{\chi}]$, element $x \in X$, and parameter $T \geq 1$, we define
\[
\cN_{c,\tau}(x,T) = \# \left \{ v \in {\bf X}(\mathbb{Q}) : d(x,v) < c \, H_{\chi}(v)^{-\tau}, \, 1 \leq H_{\chi}(v) < T \right \}.
\]
In \cite{Pfitscher24}, we provided an almost-sure asymptotic formula for $\cN_{c,\tau}(x,T)$ as $T \rightarrow +\infty$, with an explicit error term in the case where $\tau \in [0, \beta_{\chi})$. Our method did not yield an effective estimate when counting \emph{at the Diophantine exponent}, that is, when $\tau = \beta_{\chi}$. In our application, we upgrade our previous result to an effective asymptotic estimate. Our approach is inspired by a recent effective counting result due to Ouaggag \cite[Theorem~1.2]{Ouaggag23} for spheres, and our result may be viewed as a substantial generalization thereof.

\begin{thmx}[Effective counting at the Diophantine exponent] \label{thm:Critical}
Let $d = \dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{eq:Thm_Critical}
\mathcal{N}_{c, \beta_{\chi}}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}
\end{thmx}

\subsection{Proof sketch of Theorem \ref{thm:Critical}}
Let us illustrate Theorem~\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \mathrm{Gr}_{\ell, n}(\mathbb{R})$ ($\ell, n \in \mathbb{N}, \, 1 \leq \ell < n$), parametrizing $\ell$-dimensional subspaces of the Euclidean space $\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.

Let ${\bf G} = \SL_n$, let ${\bf T} \leq {\bf G}$ be the maximal $\mathbb{Q}$-split $\mathbb{Q}$-torus given by the subgroup of ${\bf G}$ consisting of all diagonal matrices, and let $\bP_0$ be the Borel subgroup of ${\bf G}$ consisting of all upper-triangular matrices. Let $\Phi = \Phi({\bf G}, {\bf T})$ be the associated root system with ordering induced by $\bP_0$, $\Delta = \{\alpha_1, \dots, \alpha_{n-1}\}$ the set of simple roots, and $\{\lambda_{1}, \dots, \lambda_{n-1} \}$ the set of fundamental $\mathbb{Q}$-weights. Fix $\alpha_{\ell} \in \Delta$ and let $\chi = \lambda_{\ell}$ be the associated fundamental $\mathbb{Q}$-weight. Recall that for all $a = \diag(a_1, \dots, a_n) \in {\bf T}$, we have $\chi(a) = a_1 \cdots a_{\ell}$. Let ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha_{\ell}\}}$ be the corresponding standard parabolic $\mathbb{Q}$-subgroup. Then ${\bf P}$ is the stabilizer in ${\bf G}$ of the rational line spanned by the pure tensor $\bm{e}_{\chi} = \bm{e}_1 \wedge \dots \wedge \bm{e}_{\ell}$ in the $\ell$-th exterior power of the standard representation of ${\bf G}$. The Siegel transform in this case is defined as follows. Let $\widetilde{X} = G \, \bm{e}_{\chi} \subset \bigwedge^\ell \mathbb{R}^n$: this is the set of all non-zero pure tensors of $\bigwedge^\ell \mathbb{R}^n$. Let ${\bf L} = \mathrm{Stab}_{{\bf G}} ( \bm{e}_{\chi})$, $\Gamma = \SL_n(\mathbb{Z})$ and let $\cP_{\chi}$ be the set of all primitive elements of $\bigwedge^\ell \mathbb{Z}^n$ that are contained in $\widetilde{X}$. The group $\Gamma$ acts transitively on $\cP_{\chi}$: $\cP_{\chi} = \Gamma \, \bm{e}_{\chi} \cong \Gamma/\G_L$. Therefore, for every $f \in B_c^{\infty}(\widetilde{X})$, the Siegel transform $S_{\chi} f : G / \Gamma \rightarrow \mathbb{C}$ is given by
\[
\forall \, g \in G, \qquad S_{\chi} f(g \Gamma) = \sum_{\bm{v} \in \cP_{\chi}} f(g \bm{v}) = \sum_{\gamma \in \Gamma / \G_L} f(g \gamma \bm{e}_{\chi}).
\]
Then $X = G/P$, viewed as a subvariety of $\mathbb{P}(\bigwedge^\ell \mathbb{R}^n)$ via the embedding $g P \mapsto g [\bm{e}_{\chi}]$ (here $[\bm{e}_{\chi}]$ denotes the projectivization of $\bm{e}_{\chi}$), is the Grassmann variety $\mathrm{Gr}_{\ell,n}(\mathbb{R})$ of $\ell$-dimensional subspaces of $\mathbb{R}^n$. This is in accordance with Schmidt's paper \cite{Schmidt67}, where he used the Pl\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\mathbb{R}^n$: for $v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q})$ pick $\bm{v} \in \cP_{\chi}$ with $v = [\bm{v}]$ and set $H(v) = \|\bm{v}\|$, where $\|\cdot \|$ denotes the $\SO_n(\mathbb{R})$-invariant norm on $\bigwedge^{\ell} \mathbb{R}^n$ induced from the standard Euclidean norm on $\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\sigma_X$ invariant under the action of the maximal compact subgroup $K = \SO_n(\mathbb{R}) \leq G$. We study the approximation of a real subspace chosen randomly according to $\sigma_X$ by rational subspaces. Write $d$ for the dimension $\dim_\mathbb{R} X = \ell(n-\ell)$. The Diophantine exponent of $X = \mathrm{Gr}_{\ell,n}(\mathbb{R})$ with respect to $\chi = \lambda_{\ell}$ is given by $\beta_{\chi} = \frac{n}{\ell(n-\ell)}$ (see \cite[Th\'eor\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function
\begin{equation} \label{eq:CountingFunction}
\cN_{c,\beta_{\chi}}(x,T) = \# \left \{ v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q}) : d(x,v) < c \, H(v)^{-\beta_{\chi}}, \, H(v) < T \right \}
\end{equation}
as $T \rightarrow + \infty$, for $\sigma_X$-almost every $x \in X$. In fact, Theorem \ref{thm:Critical} takes the following form in this special case.

Let $\psi : \mathbb{N} \rightarrow (0,+\infty)$ be a non-increasing function. By Khintchine's theorem~\cite{Khintchine26}, the inequality
\[
0 \leq q x - p < \psi(q)
\]
has infinitely (resp. at most finitely) many solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ for almost every $x \in \mathbb{R}$, if the series $\sum_{q = 1}^{\infty} \psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \in \mathbb{R}$ and $T \geq 1$, he considered the counting function
\begin{equation} \label{eq:SchmidtCountingFunction}
\cN_\psi(x,T) = \# \left \{ (p,q) \in \mathbb{Z} \times \mathbb{N} : 0 \leq q x - p < \psi(q), \, 1 \leq q < T \right \}
\end{equation}
and showed that for almost every $x \in \mathbb{R}$, $\cN_\psi(x,T)$ is asymptotically equal to $\sum_{1\leq q < T} \psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\mathbb{R}^n$ of any dimension $n \geq 1$.

\begin{thmx}[Effective counting at the Diophantine exponent] \label{thm:Critical}
Let $d = \dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{eq:Thm_Critical}
\mathcal{N}_{c, \beta_{\chi}}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}
\end{thmx}

\subsection{Proof sketch of Theorem \ref{thm:Critical}}
Let us illustrate Theorem~\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \mathrm{Gr}_{\ell, n}(\mathbb{R})$ ($\ell, n \in \mathbb{N}, \, 1 \leq \ell < n$), parametrizing $\ell$-dimensional subspaces of the Euclidean space $\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.

Let ${\bf G} = \SL_n$, let ${\bf T} \leq {\bf G}$ be the maximal $\mathbb{Q}$-split $\mathbb{Q}$-torus given by the subgroup of ${\bf G}$ consisting of all diagonal matrices, and let $\bP_0$ be the Borel subgroup of ${\bf G}$ consisting of all upper-triangular matrices. Let $\Phi = \Phi({\bf G}, {\bf T})$ be the associated root system with ordering induced by $\bP_0$, $\Delta = \{\alpha_1, \dots, \alpha_{n-1}\}$ the set of simple roots, and $\{\lambda_{1}, \dots, \lambda_{n-1} \}$ the set of fundamental $\mathbb{Q}$-weights. Fix $\alpha_{\ell} \in \Delta$ and let $\chi = \lambda_{\ell}$ be the associated fundamental $\mathbb{Q}$-weight. Recall that for all $a = \diag(a_1, \dots, a_n) \in {\bf T}$, we have $\chi(a) = a_1 \cdots a_{\ell}$. Let ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha_{\ell}\}}$ be the corresponding standard parabolic $\mathbb{Q}$-subgroup. Then ${\bf P}$ is the stabilizer in ${\bf G}$ of the rational line spanned by the pure tensor $\bm{e}_{\chi} = \bm{e}_1 \wedge \dots \wedge \bm{e}_{\ell}$ in the $\ell$-th exterior power of the standard representation of ${\bf G}$. The Siegel transform in this case is defined as follows. Let $\widetilde{X} = G \, \bm{e}_{\chi} \subset \bigwedge^\ell \mathbb{R}^n$: this is the set of all non-zero pure tensors of $\bigwedge^\ell \mathbb{R}^n$. Let ${\bf L} = \mathrm{Stab}_{{\bf G}} ( \bm{e}_{\chi})$, $\Gamma = \SL_n(\mathbb{Z})$ and let $\cP_{\chi}$ be the set of all primitive elements of $\bigwedge^\ell \mathbb{Z}^n$ that are contained in $\widetilde{X}$. The group $\Gamma$ acts transitively on $\cP_{\chi}$: $\cP_{\chi} = \Gamma \, \bm{e}_{\chi} \cong \Gamma/\G_L$. Therefore, for every $f \in B_c^{\infty}(\widetilde{X})$, the Siegel transform $S_{\chi} f : G / \Gamma \rightarrow \mathbb{C}$ is given by
\[
\forall \, g \in G, \qquad S_{\chi} f(g \Gamma) = \sum_{\bm{v} \in \cP_{\chi}} f(g \bm{v}) = \sum_{\gamma \in \Gamma / \G_L} f(g \gamma \bm{e}_{\chi}).
\]
Then $X = G/P$, viewed as a subvariety of $\mathbb{P}(\bigwedge^\ell \mathbb{R}^n)$ via the embedding $g P \mapsto g [\bm{e}_{\chi}]$ (here $[\bm{e}_{\chi}]$ denotes the projectivization of $\bm{e}_{\chi}$), is the Grassmann variety $\mathrm{Gr}_{\ell,n}(\mathbb{R})$ of $\ell$-dimensional subspaces of $\mathbb{R}^n$. This is in accordance with Schmidt's paper \cite{Schmidt67}, where he used the Pl\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\mathbb{R}^n$: for $v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q})$ pick $\bm{v} \in \cP_{\chi}$ with $v = [\bm{v}]$ and set $H(v) = \|\bm{v}\|$, where $\|\cdot \|$ denotes the $\SO_n(\mathbb{R})$-invariant norm on $\bigwedge^{\ell} \mathbb{R}^n$ induced from the standard Euclidean norm on $\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\sigma_X$ invariant under the action of the maximal compact subgroup $K = \SO_n(\mathbb{R}) \leq G$. We study the approximation of a real subspace chosen randomly according to $\sigma_X$ by rational subspaces. Write $d$ for the dimension $\dim_\mathbb{R} X = \ell(n-\ell)$. The Diophantine exponent of $X = \mathrm{Gr}_{\ell,n}(\mathbb{R})$ with respect to $\chi = \lambda_{\ell}$ is given by $\beta_{\chi} = \frac{n}{\ell(n-\ell)}$ (see \cite[Th\'eor\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function
\begin{equation} \label{eq:CountingFunction}
\cN_{c,\beta_{\chi}}(x,T) = \# \left \{ v \in \mathrm{Gr}_{\ell,n}(\mathbb{Q}) : d(x,v) < c \, H(v)^{-\beta_{\chi}}, \, H(v) < T \right \}
\end{equation}
as $T \rightarrow + \infty$, for $\sigma_X$-almost every $x \in X$. In fact, Theorem \ref{thm:Critical} takes the following form in this special case.

\begin{thmx}[Effective counting at the Diophantine exponent] \label{thm:Critical}
Let $d = \dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\varkappa > 0$ and $\varepsilon > 0$ such that for $\sigma_X$-almost every $x \in X$, as $T \rightarrow + \infty$,
\begin{equation} \label{eq:Thm_Critical}
\mathcal{N}_{c, \beta_{\chi}}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}
\end{thmx}

\begin{proof}
We shall need the following consequence of the proof of \cite[Theorem~C]{Pfitscher24}. For every $T \geq 1$, consider the function
\[
\cN(T) = \# \left \{v \in \bX(\Q) : H_{\chi}(v) < T \right \}
\]
counting rational points in $X$ of height $< T$. Let $\beta_{\chi} \in \Q_{>0}$ be the Diophantine exponent of $X$ with respect to $\chi$ (see \cite[D\'efinition~2.4.1 et Th\'eor\`eme~2.4.5]{deSaxce20}) and let $d = \dim X$ be the dimension of $X$. Then, as $T \rightarrow + \infty$, we have $\cN(T) \sim T^{\beta_{\chi} d}$. Since there is a one-to-one correspondence between points in $\bX(\Q)$ and lines passing through $\cP_{\chi}$, by the definition of the height function $H_{\chi}$, we also have that, as $T \rightarrow + \infty$
\begin{equation} \label{eq:Number_Primitive_Bounded_Height}
\# \left \{\bm{v} \in \cP_{\chi} : \|\bm{v}\| < T \right \} \asymp T^{\beta_{\chi} d}.
\end{equation}
Fix $f \in B_{c}^{\infty}(\widetilde{X})$ and pick $r = r(\supp(f)) \geq 1$ such that $\supp(f)$ is contained in $B_{\widetilde{X}}(r) = \{\bm{v} \in \widetilde{X} : \|\bm{v}\| < r \}$. The proof now proceeds using reduction theory as presented, for instance, in \cite[Section~12, Theorem~13.1]{Borel69}. By a slight abuse of notation, we let $\mathfrak{a}$ be the Lie algebra of $T^{\circ}$ and, for every $\tau \geq 0$, let $\mathfrak{a}_{\tau} = \{Y \in \mathfrak{a} : \forall \, \beta \in \Delta, \, \beta(Y) \leq \tau \}$. We set $A_{\tau} = \exp \, \mathfrak{a}_{\tau}$ and an note that $\mathfrak{a}^- = \mathfrak{a}_0$ is the negative Weyl chamber of $\mathfrak{a}$ with respect to $\Delta$. Let $\bM_0$ be the largest $\Q$-anisotropic $\Q$-subgroup of the centralizer $\cZ_{\bG}(\bT)^{\circ}$ in $\bG$ of $\bT$ and let $\bU_0$ be the unipotent radical of the minimal parabolic $\Q$-subgroup $\bP_0$. There exist $\tau > 0$, a compact subset $\bm{\omega}$ of $M_0 U_0$, and a finite subset $C \subset \bG(\Q)$ such that the Siegel set $\mathfrak{S} = K \, A_{\tau} \, \bm{\omega}$ satisfies
\[
G = \mathfrak{S} \, C \, \G.
\]
In particular, we can express, though not uniquely, each $g \in G$ as $g = k a n c \gamma$ with $k \in K$, $a \in A_{\tau}$, $n \in \bm{\omega}$, $c \in C$, and $\gamma \in \G$. Fix any norm $\|\cdot \|_{\mathfrak{a}}$ on $\mathfrak{a}$ and, for $r_0 > 0$, let $B_{\mathfrak{a}}(r_0)$ denote the corresponding ball centered at the origin with radius $r_0$. Let $r_0 > 0$ be such that $\mathfrak{a}_{\tau}$ is contained in $\mathfrak{a}^- + B_{\mathfrak{a}}(r_0)$. Let $k \in K$, $n \in \bm{\omega}$, $a \in A_{\tau}$, $c \in C$, and $\gamma \in \G$. We express $a = a^- \, \exp(O(1))$ with $a^- \in \exp(\mathfrak{a}^-)$. Using that $\lambda_\chi$ is right $\G$-invariant, that $K$ is compact, that $\bigcup_{a \in A_{\tau}} a \bm{\omega}a^{-1}$ is relatively compact (see \cite[Lemma~12.2]{Borel69}), and that $C \subset \bG(\Q)$ consists of rational elements and is finite, we have
\begin{equation} \label{eq:Proof-lem:Upper-Bound-Siegel}
\lambda_\chi(k a n c \gamma \G) \asymp \lambda_\chi(a^- \G).
\end{equation}
By the description of the $\Q$-weights of the representation $\pi_{\chi}$ in \eqref{eq:Q-weights}, for every $\Q$-weight $\mu$ of $\pi_{\chi}$, we have
\[
\chi(a^-) \leq \mu(a^-).
\]
Hence, since we assumed $\bV_{\chi}(\Z)$ to be spanned over $\Z$ by an orthonormal basis consisting of weight vectors for the action of $T$, we have $\lambda_\chi(a^-) = \chi(a^-)$. Thus, for every $\bm{v} \in V_{\chi}$, we have $\lambda_\chi(a^-) \| \bm{v} \| \leq \|a^- \bm{v} \|$. Using that the norm $\| \cdot \|$ on $V_{\chi}$ is $K$-invariant, that $\bigcup_{a \in A_{\tau}} a \bm{\omega}a^{-1}$ is relatively compact, and that $C \subset \bG(\Q)$ is finite, there exists a constant $C_0 \geq 1$, independent of $f$, such that, for every $g \in G$ with Siegel decomposition $g = k a n c \gamma$ (and writing $a = a^- \exp(O(1))$ as above), we have
\begin{align*}
|S_{\chi} f(g \G)| &\leq \|f\|_{\infty} \, \# \big \{ \bm{v} \in \cP_{\chi} : \| g \bm{v} \| < r \big \} \\
&\leq \|f\|_{\infty} \, \# \big \{ \bm{v} \in \cP_{\chi} : \| \bm{v} \| < C_0 \, \lambda_\chi(a^-)^{-1} \, r \big \}.
\end{align*}
By the estimate in \eqref{eq:Number_Primitive_Bounded_Height}, we further have
\[
\# \big \{ \bm{v} \in \cP_{\chi} : \| \bm{v} \| < C_0 \, \lambda_\chi(a^-)^{-1} \, r \big \} \, \ll_{\supp(f)} \, \lambda_\chi(a^-)^{-\beta_{\chi} d}.
\]
This together with \eqref{eq:Proof-lem:Upper-Bound-Siegel} now implies that
\[
|S_{\chi} f(g\G)| \ll_{\supp(f)} \|f\|_{\infty} \, \lambda_\chi(g \G)^{-\beta_\chi d},
\]
finishing the proof of the lemma.
\end{proof}

Let us now define these sets that approximate $\cE_{\beta_{\chi}}(T)$. We recall from Section \ref{sec:Distance} that the map $\ku^- \rightarrow X$ sending $u \mapsto \exp(u) x_0$ restricts to a diffeomorphism from a neighborhood of $1 \in \ku^-$ to a neighborhood of $x_0 \in X$. In particular, any $\bm{v} \in \widetilde{X}$, such that $[\bm{v}]$ is close to $x_0$, defines an element $u_{\bm{v}}^-$ in the Lie algebra $\ku^-$ by $[\bm{v}] = \exp(u_{\bm{v}}^-) x_0$. The adjoint action of $a(y) \in A$ on $\ku^- = T_{x_0} X$ acts by scalar multiplication: for all $y \in \R_+^\times$, $\Ad(a(y)) u^- = y \, u^-$. Observe that
\[
[a(y) \bm{v}] = a(y) [\bm{v}] = a(y) \exp(u_{\bm{v}}^-) a(y) a(y)^{-1} x_0 = \exp(\Ad(a(y)) u_{\bm{v}}^-) x_0.
\]
But we also have $[a(y) \bm{v}] = \exp(u_{a(y) \bm{v}}^- ) x_0$. By uniqueness, this gives the relation
\begin{equation} \label{eq:LieAlgebraDiagonal}
u_{a(y) \bm{v}}^- = y \, u_{\bm{v}}^-.
\end{equation}
Moreover, by the distance estimate \eqref{eq:Distance_Estimate}, there exists a constant $C_0 > 0$ such that
\[
\left | d(x_0, [\bm{v}]) - \|u_{\bm{v}}^-\|_{\ku^-} \right | \leq C_0 \, \|u_{\bm{v}}^-\|_{\ku^-}^2.
\]
Let $\pi^+ : V_\chi \rightarrow V_\chi$ be the orthogonal projection onto $\R\bm{e}_{\chi}$ and we abbreviate $\pi^+(\bm{v})$ by $\bm{v}^+$. For every $T \geq 1$ and $c > 0$ close to $1$, we will work with the sets
\[
\cE_{T, c}^+ = \{\bm{v} \in \widetilde{X} : \|u_{\bm{v}}^-\|_{\ku^-} < c \, \|\bm{v}^+\|^{-\beta_{\chi}}, 1 \leq \|\bm{v}^+\| < c \, T \}.
\]
By enlarging $C_0$ if necessary, we can assume that $\|\bm{v}^+\| \geq C_0^{-1} \|\bm{v}\|$ as soon as $d(x_0, [\bm{v}]) < 1$. For every integer $\ell \geq 1$, we let
\[
Q_{\ell} = \{\bm{v} \in \widetilde{X} : \|\bm{v}\| \leq C_0 \, \ell \}
\]
and we define
\[
\widehat{c}_{\ell} = \left ( 1 + C_0 \, \ell^{-\beta_{\chi}} \right )^{-(1+\beta_{\chi})} \in (0,1).
\]
In particular, we have $\widehat{c}_{\ell} \nearrow 1$ as $\ell \rightarrow + \infty$.

Let us now go through the main steps of the argument. For simplicity, we assume that $c = 1$ and we write $\cN_{\beta_{\chi}}(x,T) = \cN_{1,\beta_{\chi}}(x,T)$. The first observation is that the quantity $\cN_{\beta_{\chi}}(x,T)$ can be understood as the Siegel transform of the indicator function of a certain subset $\cE_{\beta_{\chi}}(T) \subset \widetilde{X}$ evaluated at a certain point in $\Omega = G/\Gamma$: we can associate to each $x \in X$ an element $k_{x} \in K$ such that
\[
\cN_{\beta_{\chi}}(x,T) = \# (\cP_{\chi} \cap k_{x} \, \cE_{\beta_{\chi}}(T)) = S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)} (k_{x}^{-1} \Gamma).
\]
By Theorem \ref{thm:L1}, since ${\bf P} = \bP_{\Delta \smallsetminus \{\alpha_{\ell}\}}$ is maximal, the group $L = {\bf L}(\mathbb{R})$ is unimodular, $\widetilde{X} = G / L$ admits a unique up to scaling Radon measure $\lambda_{\widetilde{X}}$ and the expected value of $S_{\chi} \mathbbm{1}_{\cE_{\beta}(T)}$, viewed as a random variable on $\Omega$, is given by
\[
\int_{\Omega} S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)} \, \,\mathrm{d} \mu_{\Omega} = \int_{\widetilde{X}} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)} \, \,\mathrm{d} \lambda_{\widetilde{X}} = \lambda_{\widetilde{X}} (\cE_{\beta_{\chi}}(T)).
\]
The hope is that, for $\sigma_X$-almost every $x \in X$, the quantity $\cN_{\beta_{\chi}}(x,T)$ is asymptotically equal to the volume $\lambda_{\widetilde{X}} (\cE_{\beta_{\chi}}(T))$, as $T \rightarrow + \infty$, and this is what we will show. In fact, the main term on the right-hand side in \eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\lambda_{\widetilde{X}} (\cE_{\beta_{\chi}}(T))$. In order to prove the desired asymptotic estimate, we will exploit the special geometry of the set $\cE_{\beta_{\chi}}(T)$. In fact, this set can be approximated by a set $\cE_{\beta_{\chi}}(T)^{+}$ that admits a simple decomposition under the action of the diagonal subgroup
\[
\forall \, y \in \R_+^\times, \quad a(y) = \diag \big (\underbrace{y^{-(n-\ell)/n}, \dots, y^{-(n-\ell)/n}}_{\text{$\ell$ times}}, \underbrace{y^{\ell/n}, \dots, y^{\ell/n}}_{\text{$n-\ell$ times}} \big ).
\]
Indeed, there exists a subset ${\mathcal F} \subset \widetilde{X}$ such that for all integers $N \geq 1$
\begin{equation} \label{eq:Proof_Sketch_Decomposition}
\cE_{\beta_{\chi}}(e^N)^+ = \bigsqcup_{i=0}^{N-1} a(e^{\beta_{\chi}})^{-i} {\mathcal F}.
\end{equation}

\begin{figure}[htbp]
\includegraphics[scale=0.6]{figure_counting.png}
\caption{The set $\cE_{\beta_{\chi}}(T)^+$ for the group $G = \SL_2(\mathbb{R})$, the flag variety the real projective line $X = \mathbb{P}^1(\mathbb{R}) = \mathrm{Gr}_{1,2}(\mathbb{R})$, the punctured affine cone $\widetilde{X} = \mathbb{R}^2 \setminus \{0\}$ above $X$, and the set $\cP_{\chi} = \mathcal{P}(\mathbb{Z}^2)$ of primitive elements of $\mathbb{Z}^2$. Rational approximations to a point $x \in X$ of height bounded by $T$ correspond to primitive lattice points of $\mathbb{Z}^2$ in the red region $k_x \cE_{\beta_{\chi}}(T)$, where $k_x \in \SO_2(\mathbb{R})$ is a rotation such that $x = k_x [\bm{e}_1]$. The action of $a(y) = \diag ( y^{-1/2}, y^{1/2})$ with $y > 1$ on $\widetilde{X} = \mathbb{R}^2 \setminus \{0\}$ contracts the line through $\bm{e}_\chi = \bm{e}_1$ and expands the line through $\bm{e}_2$. The domain $\cE_{\beta_{\chi}}(T)^+$ can be decomposed into translates of the elementary domain ${\mathcal F}$ under the action of $a(y)$. The hope is that for $x$ chosen randomly according to the Lebesgue measure on $X$ the number of primitive lattice points in the red region $k_x \cE_{\beta_{\chi}}(T)$, that is, the quantity $\# (\cP_{\chi} \cap k_x \, \cE_{\beta_{\chi}}(T))$, which is the classical primitive Siegel transform of the indicator function $\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}$ evaluated at the rotated lattice $k_x^{-1} \mathbb{Z}^2$, is approximately given (up to a scalar) by the volume of $\cE_{\beta_{\chi}}(T)$.}
\label{fig:4}
\end{figure}

On the level of the Siegel transform this yields the sum decomposition
\[
S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k_x^{-1} \Gamma) = \sum_{i=0}^{N-1} \mathbbm{1}_{{\mathcal F}}(a(e^{\beta_{\chi}})^{i} k_x^{-1} \Gamma).
\]
From now on, we simply view $S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k \Gamma)$ as a random variable on the probability space $(K, \mu_K)$, where $\mu_K$ is the Haar probability measure of $K$.
Up to dividing the right-hand side by $N$, it is a Birkhoff sum, but we will not take this viewpoint. Instead, we shall try to bound a quantity related to the variance of $S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}$ and then conclude by a Borel-Cantelli argument. More specifically, we shall bound a $(1+\varepsilon)$-moment, for some $\varepsilon > 0$, of the centered Siegel transform $S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+} - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T)^+)$, viewed as a random variable on $(K,\mu_K)$: if we can show that for some $\varepsilon > 0$ and all $N \geq 1$,
\begin{equation} \label{eq:Proof_Sketch}
\int_K \left | S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma) - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+) \right |^{1+\varepsilon} \, \,\mathrm{d} \mu_{K}(k) \, \ll \, N,
\end{equation}
then there exists $c > 0$ and $\nu(\varepsilon) \in (0,1)$ such that for $\mu_K$-almost every $k \in K$,
\[
S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma) = c \, N \, \left ( 1 + O_x(N^{-\nu(\varepsilon)})\right ),
\]
as required. Due to integrability issues of the Siegel transform at this level of generality (see Theorems \ref{thm:L1} and \ref{thm:L2}), we are forced to work with $1 + \varepsilon$ for some small $\varepsilon > 0$ instead of $2$, which would represent the usual variance.
Using the decomposition \eqref{eq:Proof_Sketch_Decomposition}, we express the argument in the integral of \eqref{eq:Proof_Sketch} as
\begin{equation*}
S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma) - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+)
= \sum_{i=0}^{N-1} \bigg ( S_{\chi} \mathbbm{1}_{{\mathcal F}} (a(e^{\beta_{\chi}}) k_x^{-1} \Gamma) - \lambda_{\widetilde{X}}({\mathcal F}) \bigg ).
\end{equation*}
and obtain the bound in \eqref{eq:Proof_Sketch} using the effective single and double equidistribution property of expanding translates of $K$-orbits. In particular, we will need to work with smooth compactly supported functions that, on translated $K$-orbits, approximate the Siegel transform $S_{\chi} \mathbbm{1}_{{\mathcal F}}$, which typically is neither smooth nor compactly supported.

\subsection{Notation and conventions}
We use the Landau notation $O(\cdot)$ and the Vinogradov symbol $\ll$. Given $A, B > 0$, we use the notation $A \gg B$ for $B \ll A$, and $A \asymp B$ for $A \ll B \ll A$. We use subscripts to indicate the dependence of the constant on parameters.
For simplicity of exposition, we will work with the set of complex points of an algebraic variety defined over $\mathbb{Q}$, and refer to it simply as the variety itself when no confusion arises. For instance, we write $G = {\bf G}(\mathbb{R})$ and ${\bf G} = {\bf G}(\mathbb{C})$ to denote the groups of real and complex points of ${\bf G}$, respectively. Given a discrete subgroup $\Gamma \leq G$ and a closed subgroup $H \leq G$, we write $\G_H$ for $\Gamma \cap H$. Discrete groups are always equipped with the counting measure.

\vspace{5mm}
\textbf{Acknowledgments}.
I am very grateful to Nicolas de Saxc\'e for introducing me to this topic, during my doctoral thesis under his supervision, and for sharing with me crucial insights that contributed to the proofs of Theorems \ref{thm:L2} and \ref{thm:Linfty}. I also thank Shucheng Yu for useful discussions, and Fr\'ed\'eric Paulin for numerous corrections and suggestions that led, in particular, to the removal of a restrictive hypothesis in Theorem \ref{thm:Critical} (namely, that $\beta_{\chi} \leq 1$).

Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\cN_{\beta_{\chi}}(x,T)$ as a Siegel transform: for $x\in X$ one associates $k_x\in K$ such that
\[
\cN_{\beta_{\chi}}(x,T)=\#(\cP_{\chi}\cap k_x\cE_{\beta_{\chi}}(T))=S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}(k_x^{-1}\Gamma).
\]
By Theorem \ref{thm:L1}, the expected value (on $\Omega$) of this random variable is
\[
\int_{\Omega} S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\mu_{\Omega}=\int_{\widetilde{X}}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\lambda_{\widetilde{X}}=\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T)).
\]
The goal is to show that for $\sigma_X$-almost every $x$, $\cN_{\beta_{\chi}}(x,T)$ is asymptotically equal to this volume as $T\to+\infty$; moreover, “the main term on the right-hand side in \eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T))$.”

To get an effective almost-everywhere estimate, one exploits the geometry of $\cE_{\beta_{\chi}}(T)$ by approximating it with a set $\cE_{\beta_{\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\mathcal F}\subset\widetilde{X}$ such that for integers $N\ge1$,
\[
\cE_{\beta_{\chi}}(e^N)^+=\bigsqcup_{i=0}^{N-1} a(e^{\beta_{\chi}})^{-i}{\mathcal F}. \tag{\ref{eq:Proof_Sketch_Decomposition}}
\]
On the Siegel-transform level this gives
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k_x^{-1}\Gamma)=\sum_{i=0}^{N-1}\mathbbm{1}_{{\mathcal F}}(a(e^{\beta_{\chi}})^i k_x^{-1}\Gamma),
\]
and one views $S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k\Gamma)$ as a random variable on $(K,\mu_K)$.

Rather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\varepsilon>0$ and all $N\ge1$,
\[
\int_K\Big|S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma)-\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+)\Big|^{1+\varepsilon}\,d\mu_K(k)\ll N, \tag{\ref{eq:Proof_Sketch}}
\]
then one deduces that for $\mu_K$-almost every $k\in K$,
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma)=c\,N\,(1+O_x(N^{-\nu(\varepsilon)})).
\]
Because of “integrability issues of the Siegel transform … we are forced to work with $1+\varepsilon$ … instead of $2$.” Using the decomposition \eqref{eq:Proof_Sketch_Decomposition}, the centered sum is written as a sum of $N$ terms involving $S_{\chi}\mathbbm{1}_{\mathcal F}$, and the bound \eqref{eq:Proof_Sketch} is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\chi}\mathbbm{1}_{\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.

Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\cN_{\beta_{\chi}}(x,T)$ as a Siegel transform: for $x\in X$ one associates $k_x\in K$ such that
\[
\cN_{\beta_{\chi}}(x,T)=\#(\cP_{\chi}\cap k_x\cE_{\beta_{\chi}}(T))=S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}(k_x^{-1}\Gamma).
\]
We first recall the following theorem.
\begin{thmx} [$L^1$-integrability] \label{thm:L1}
The following assertions are equivalent.
\begin{enumerate}[label=(\arabic*)]
\item The Siegel transform $S_\chi$ maps $B_c^{\infty}(\widetilde{X})$ into $L^1(\Omega)$.
\item There exists a unique (up to scaling) $G$-invariant Radon measure $\lambda_{\widetilde{X}}$ on $\widetilde{X}$, the Siegel transform $S_\chi$ extends to a bounded operator $S_{\chi} : L^1(\widetilde{X}) \rightarrow L^1(\Omega)$, and $\lambda_{\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:
\begin{equation} \label{eq:WeilIntegrationFormula*}
\forall \, f \in L^1(\widetilde{X}), \quad \int_{\Omega} S_\chi f \, \dd \mu_{\Omega} = \int_{\widetilde{X}} f \, \dd \lambda_{\widetilde{X}}.
\end{equation}
\item The Lie group $L = \bL(\R)$ is unimodular and $\G_L = \G \cap L$ is a lattice in $L$.
\item The parabolic $\Q$-subgroup $\bP$ of $\bG$ is maximal.
\item There exists $\varepsilon > 0$ such that $S_\chi$ maps $B_{c}^{\infty}(\widetilde{X})$ into $L^{1+\varepsilon}(\Omega)$.
\end{enumerate}
\end{thmx}
By this theorem, the expected value (on $\Omega$) of this random variable is
\[
\int_{\Omega} S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\mu_{\Omega}=\int_{\widetilde{X}}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)}\,d\lambda_{\widetilde{X}}=\lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(T)).
\]
The goal is to show that for $\sigma_X$-almost every $x$, $\cN_{\beta_{\chi}}(x,T)$ is asymptotically equal to this volume as $T\to+\infty$; moreover, the main term in the volume asymptotic is given by the explicit formula
\begin{equation} \label{cor:eq_Grassmannian}
\mathcal{N}_{c, \beta}(x, T) = \varkappa \, c^d \, \ln (T) \left ( 1 + O_x(\ln(T)^{- \varepsilon}) \right ).
\end{equation}

To get an effective almost-everywhere estimate, one exploits the geometry of $\cE_{\beta_{\chi}}(T)$ by approximating it with a set $\cE_{\beta_{\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\mathcal F}\subset\widetilde{X}$ such that for integers $N\ge1$,
\begin{equation} \label{eq:Proof_Sketch_Decomposition}
\cE_{\beta_{\chi}}(e^N)^+ = \bigsqcup_{i=0}^{N-1} a(e^{\beta_{\chi}})^{-i} \cF.
\end{equation}
On the Siegel-transform level this gives
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k_x^{-1}\Gamma)=\sum_{i=0}^{N-1}\mathbbm{1}_{{\mathcal F}}(a(e^{\beta_{\chi}})^i k_x^{-1}\Gamma),
\]
and one views $S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(T)^+}(k\Gamma)$ as a random variable on $(K,\mu_K)$.

Rather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\varepsilon>0$ and all $N\ge1$,
\begin{equation} \label{eq:Proof_Sketch}
\int_K \left | S_{\chi} \mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\G) - \lambda_{\widetilde{X}}(\cE_{\beta_{\chi}}(e^N)^+) \right |^{1+\varepsilon} \, \dd \mu_{K}(k) \, \ll \, N,
\end{equation}
then one deduces that for $\mu_K$-almost every $k\in K$,
\[
S_{\chi}\mathbbm{1}_{\cE_{\beta_{\chi}}(e^N)^+}(k\Gamma)=c\,N\,(1+O_x(N^{-\nu(\varepsilon)})).
\]
Because of “integrability issues of the Siegel transform … we are forced to work with $1+\varepsilon$ … instead of $2$.” Using the decomposition above, the centered sum is written as a sum of $N$ terms involving $S_{\chi}\mathbbm{1}_{\mathcal F}$, and the bound above is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\chi}\mathbbm{1}_{\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.

Asymptotic or Limit
Existential–Universal

Question: Let integers \(1\le \ell<n\) be fixed, and let \(X=\mathrm{Gr}_{\ell,n}(\mathbb{R})\) be the Grassmannian of \(\ell\)-dimensional subspaces of \(\mathbb{R}^n\), equipped with its natural measure \(\sigma_X\). Let \(d=\dim X\), fix \(c>0\), and for \(x\in X\) and \(T\ge 1\), let \(\mathcal N_{c,\beta}(x,T)\) denote the counting function for rational \(\ell\)-planes \(v\in \mathrm{Gr}_{\ell,n}(\mathbb{Q})\) with height \(H(v)<T\) and distance from \(x\) bounded by \(cH(v)^{-\beta}\), where \(H(v)\) is the Pl\"ucker height (the Euclidean norm of a primitive Pl\"ucker vector representing \(v\)). Which asymptotic statement holds for \(\mathcal N_{c,\beta}(x,T)\) as \(T\to+\infty\)?

There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for \(\sigma_X\)-almost every \(x\in X\), as \(T\to+\infty\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c^d\,\ln(T)\left(1+O_x\big((\ln T)^{-\varepsilon}\big)\right).
\]

Choices:
B.
There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for every \(x\in X\), as \(T\to+\infty\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c^d\,\ln(T)\left(1+O\big((\ln T)^{-\varepsilon}\big)\right).
\]
C.
There exists an explicit constant \(\varkappa>0\) such that for \(\sigma_X\)-almost every \(x\in X\), as \(T\to+\infty\),
\[
\mathcal N_{c,\beta}(x,T)\sim \varkappa\,c^d\,\ln(T).
\]
D.
There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for \(\sigma_X\)-almost every \(x\in X\), as \(T\to+\infty\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c^d\,T\left(1+O_x\big(T^{-\varepsilon}\big)\right).
\]
E.
There exist explicit constants \(\varkappa>0\) and \(\varepsilon>0\) such that for \(\sigma_X\)-almost every \(x\in X\), as \(T\to+\infty\),
\[
\mathcal N_{c,\beta}(x,T)=\varkappa\,c\,\ln(T)\left(1+O_x\big((\ln T)^{-\varepsilon}\big)\right).
\]

The stem does not explicitly reveal the correct option or quote the conclusion verbatim; it mainly sets up notation and asks for the asymptotic. There is no direct answer leakage.

This is essentially a theorem-recall item: after stating the full hypotheses, it asks which asymptotic holds. The correct choice is basically the theorem's conclusion rather than a derived consequence from competing premises.

There is some pressure to distinguish subtle variants (almost every vs every, effective error term vs mere asymptotic, logarithmic vs polynomial growth, and c^d vs c), but the item primarily tests recall of a known statement rather than genuine mathematical generation from first principles.

The distractors are plausible and mathematically meaningful: B overstrengthens the quantifier, C gives a weaker true statement, D swaps in an incorrect growth rate, and E tampers with the scaling in c. These are distinct and aligned with common failure modes.

A solid theorem-discrimination MCQ with strong distractors and no obvious leakage, but it is largely a direct restatement/recall question rather than a genuinely generative reasoning task.


\label{them:classresult}
  There exists a $X$ in every deformation family of Fano threefolds such that $\operatorname{AutP}(X)=\wg_X$ except possibly for families
  \begin{center}
      \textnumero 2.2, \textnumero 3.9, \textnumero 4.2, \textnumero 6.1, \textnumero 7.1, \textnumero 8.1, \textnumero 9.1, \textnumero 10.1.
  \end{center}

\begin{thm}  \label{them:classresult}
  There exists a $X$ in every deformation family of Fano threefolds such that $\autp(X)=\wg_X$ except possibly for families
  \begin{center}
      \textnumero 2.2, \textnumero 3.9, \textnumero 4.2, \textnumero 6.1, \textnumero 7.1, \textnumero 8.1, \textnumero 9.1, \textnumero 10.1.
  \end{center}
\end{thm}

\label{imalphasection}
Let $X$ be a Fano threefold. Recall that $\autp(X,G)$ denotes the image of $G \hookrightarrow \aut(X) \xrightarrow{\alpha} \aut(\pic(X))$. In \cite[Definition 2.8]{hmonpaper}, t

\label{table}
We summarize the results from previous sections in the table below. In column 1, we write the Mori-Mukai name for the deformation family of Fano threefolds. Column 2 lists $\autp(X)$, th

\begin{thm}  \label{thm:H1main}
    Let $X$ be a smooth Fano threefold with Picard rank $\rho \leq 5$. Let $G\subseteq \aut(X)$ be a finite group. Then $H^1(G,\pic(X))=0$.
\end{thm}

\begin{center}
        \textnumero 5.1,\; \textnumero 5.2,\; \textnumero 5.3,\; \textnumero 6.1,\; \textnumero 7.1,\; \textnumero 8.1,\; \textnumero 9.1,\; \textnumero 10.1.
        \end{center}
    \end{enumerate}
\end{coro}

For each of the families listed in the above corollary, we now find whether $\autp(X)=0$ or the upper bound on it given by the Weyl group can be realized for a group $G\subseteq \aut(X)$.

\section{Classification of \texorpdfstring{$\autp(X,G)$}{AutP(X,G)}}  \label{proof:classresult}
\subsection {Divisors on a products of projective spaces}   \label{im(alpha)divisortype}
Let $X$ be a smooth Fano threefold that is a member of any of the five deformation families
\begin{center}
\textnumero 2.6\;(a),\; \textnumero 2.32,\; \textnumero 3.3,\; \textnumero 3.17, \; \textnumero 4.1.
\end{center}

Let $X$ be a smooth projective variety over $\mathbb{C}$. We say $X$ is Fano if its anticanonical divisor $-K_X$ is ample.
Iskovskikh, Mori, and Mukai have classified all deformation families of Fano threefolds, see \cite{isko1, isko2, MM81classtable, mori1983fano, MM86, MMerratum}. There are $105$ such families. A complete list of deformation families with a description of their general smooth member along with other algebraic-geometric information can be found in \cite{fanography}. The classification of Fano threefolds was recently completed in positive characteristic by Asai and Tanaka in \cite{tanaka-I, tanaka-II, asaitanaka-III, tanaka2023IV}.

The classification of automorphism groups of smooth Fano threefolds is a highly challenging problem, and efforts to understand it are ongoing. The connected components of the identity of automorphism groups of all smooth Fano threefolds were classified in \cite{hilbert, izv19}.
This identity component acts trivially on the Picard group of $X$. Thus, by studying the natural action of automorphism group on the Picard group, one may hope for a deeper understanding of the automorphism group itself. The automorphism groups of smooth cubic threefolds were classified in \cite{wei2019automorphismgroupssmoothcubic}. Several other recent works have investigated the automorphism groups of various families of Fano threefolds; see, for instance, \cite{calabi, cheltsovs2-12family, abe2025automorphismgroupslinearizabilityrational}.

Suppose that $X$ is a Fano threefold and $G$ is a finite group of automorphisms of $X$. The Leray spectral sequence of the $G$-action on $X$ gives the exact sequence (\cite[\S 3]{kresch2022cohomologyfinitesubgroupsplane})
\begin{align*}
1 \rightarrow \operatorname{Hom}(G,\mathbb{C}^\times)& \rightarrow \operatorname{Pic}(X,G) \rightarrow \operatorname{Pic}(X)^G \xrightarrow{\partial} H^2(G,\mathbb{C}^\times)\to \\
& \operatorname{Br}([X/G])\to H^1(G,\operatorname{Pic}(X))\to H^3(G,\mathbb{C}^\times).
\end{align*}
where $\operatorname{Pic}(X,G)$ and $\operatorname{Pic}(X)^G$ are the groups of isomorphism classes of $G$-linearized and $G$-invariant line bundles on $X$, respectively. $\operatorname{Br}([X/G])$ is the Brauer group of quotient stack $[X/G]$. The group $\operatorname{Am}(X,G):=\operatorname{im}(\partial)$ is called the \emph{Amitsur subgroup}. The three groups, $\operatorname{Am}(X,G)$, $H^1(G,\operatorname{Pic}(X))$ and $\operatorname{Br}([X/G])$ are important stable equivariant birational invariants (\cite{BCDP, H1isbirinvariant, hassett_tschinkel_odd}). They provide obstructions to equivariant versions of arithmetic analogs of rationality (such as linearizability) and have been a focus of recent interest in \cite{kresch2022cohomologyfinitesubgroupsplane,kresch-unramified-brauergroup,equibrauergroup,tschinkel2025cohomologicalobstructionsequivariantunirationality}.

The Amitsur subgroup is classified for smooth Fano varieties of dimension 1 and 2 over algebraically closed fields (\cite{BCDP, dolga-modular-curves}). The first cohomology group $H^1(G,\operatorname{Pic}(X))$ has also been studied for smooth del Pezzo surfaces, see \cite{manin-cubicforms, urabe-H1computation} for the arithmetic case and \cite{H1isbirinvariant,kresch2022cohomologyfinitesubgroupsplane} for an application to the plane Cremona group. In order to compute both of these groups, it is desirable to determine the induced $G$-action on $\operatorname{Pic}(X)$.

The main objective of this paper is to determine, for each family of smooth Fano threefolds $X$, the possible images $\operatorname{AutP}(X)$ of the natural action of $\operatorname{Aut}(X)$ on $\operatorname{Pic}(X)$. The image of a finite group $G \subseteq \operatorname{Aut}(X)$ under this action preserves some combinatorial information associated to the Mori cone of $X$, see \cite[2.8, 2.9]{hmonpaper}. In \cite{Prokhorov-G-FanoII}, Prokhorov lists the families of Fano threefolds for which $\operatorname{Pic}(X)^G \cong \mathbb{Z}$, where the $G$-action need not necessarily come from $X$. In a related direction, the authors in \cite[Lemma 2.13]{abban2025kstabilityfano3foldsworld} classify the families of Fano threefolds for which every extremal ray of the Mori cone is $G$-invariant. The actual computation of the image of $G$ for all families of Fano threefolds, however, is still open. In \cite{Mat95}, K. Matsuki defines the Weyl group of a Fano threefold that preserves the same data as the image of $G$ and gives their complete classification (cf. Definition \ref{def:WG}).

In the table below, we list the families of Fano threefolds with $\rho\leq 5$ that have nontrivial Weyl group, $\wg_X$. For the remaining families, $\wg_X=0$.

\begin{table}[ht]\renewcommand{\arraystretch}{1.2}
\caption{Weyl groups of Fano threefolds}  \label{weylfano}
\begin{tabularx}{0.80\textwidth}{|X|c|}
\hline
Family & $\wg_X$ \\
\hhline{|=|=|}
\textnumero 2.2, \textnumero 2.6, \textnumero 2.12, \textnumero 2.21, \textnumero 2.32, \textnumero 3.3, \textnumero 3.7, \textnumero 3.9, \textnumero 3.10, \textnumero 3.17, \textnumero 3.19, \textnumero 3.20, \textnumero 3.25, \textnumero 3.31, & $\mathbb{Z}/2\mathbb{Z}$ \\
 \textnumero 4.3, \textnumero 4.4, \textnumero 4.7, \textnumero 4.8, \textnumero 4.10, \textnumero 4.12, \textnumero 4.13, \textnumero 5.2 &  \\
\hline
\textnumero 3.1, \textnumero 3.13, \textnumero 3.27, \textnumero 4.6, \textnumero 5.1 &  $S_3$\\
\hline
\textnumero 4.2  & $(\mathbb{Z}/2\mathbb{Z})^2$\\
\hline
\textnumero 4.1    &  $S_4$ \\
\hline
\textnumero 5.3   & $\mathbb{Z}/2\mathbb{Z} \times S_3$ \\
\hline
\end{tabularx}
\end{table}

If $X$ is a Fano threefold with $\rho \geq 6$,  then $X$ is isomorphic to $\mathbb{P}^1\times S$, where $S$ is a del Pezzo surface of degree at most $5$. In this case, the Weyl group of $X$ coincides with the Weyl group of del Pezzo surface $S$.
The Weyl groups of del Pezzo surfaces of degree at most $5$ are described in \cite[\S\S 6.3–6.7]{dolgaisko_planecremonagroups}.

We observe that the group $\operatorname{AutP}(X)$ is contained in Matsuki's Weyl group (cf. Proposition \ref{imagealphainweylgroup}).
The classification in Table \ref{weylfano} then implies that $\operatorname{AutP}(X)=0$ except possibly for the families listed there. For each deformation family with nontrivial $\wg_X$, we either exhibit a smooth member in the family admitting nontrivial automorphisms that induce nontrivial automorphisms of $\operatorname{Pic}(X)$, or show that any $G \subseteq \operatorname{Aut}(X)$ acts trivially on the Picard group.

\smallskip
The main classification result is as follows.

Suppose that $X$ is a Fano threefold and $G$ is a finite group of automorphisms of $X$. The Leray spectral sequence of the $G$-action on $X$ gives the exact sequence (\cite[\S 3]{kresch2022cohomologyfinitesubgroupsplane})
\begin{align*}
1 \rightarrow \operatorname{Hom}(G,\mathbb{C}^\times)& \rightarrow \operatorname{Pic}(X,G) \rightarrow \operatorname{Pic}(X)^G \xrightarrow{\partial} H^2(G,\mathbb{C}^\times)\to \\
& \operatorname{Br}([X/G])\to H^1(G,\operatorname{Pic}(X))\to H^3(G,\mathbb{C}^\times).
\end{align*}
where $\operatorname{Pic}(X,G)$ and $\operatorname{Pic}(X)^G$ are the groups of isomorphism classes of $G$-linearized and $G$-invariant line bundles on $X$, respectively. $\operatorname{Br}([X/G])$ is the Brauer group of quotient stack $[X/G]$. The group $\operatorname{Am}(X,G):=\operatorname{im}(\partial)$ is called the \emph{Amitsur subgroup}. The three groups, $\operatorname{Am}(X,G)$, $H^1(G,\operatorname{Pic}(X))$ and $\operatorname{Br}([X/G])$ are important stable equivariant birational invariants (\cite{BCDP, H1isbirinvariant, hassett_tschinkel_odd}). They provide obstructions to equivariant versions of arithmetic analogs of rationality (such as linearizability) and have been a focus of recent interest in \cite{kresch2022cohomologyfinitesubgroupsplane,kresch-unramified-brauergroup,equibrauergroup,tschinkel2025cohomologicalobstructionsequivariantunirationality}.

The main objective of this paper is to determine, for each family of smooth Fano threefolds $X$, the possible images $\operatorname{AutP}(X)$ of the natural action of $\operatorname{Aut}(X)$ on $\operatorname{Pic}(X)$. The image of a finite group $G \subseteq \operatorname{Aut}(X)$ under this action preserves some combinatorial information associated to the Mori cone of $X$, see \cite[2.8, 2.9]{hmonpaper}. In \cite{Prokhorov-G-FanoII}, Prokhorov lists the families of Fano threefolds for which $\operatorname{Pic}(X)^G \cong \mathbb{Z}$, where the $G$-action need not necessarily come from $X$. In a related direction, the authors in \cite[Lemma 2.13]{abban2025kstabilityfano3foldsworld} classify the families of Fano threefolds for which every extremal ray of the Mori cone is $G$-invariant. The actual computation of the image of $G$ for all families of Fano threefolds, however, is still open. In \cite{Mat95}, K. Matsuki defines the Weyl group of a Fano threefold that preserves the same data as the image of $G$ and gives their complete classification (cf. Definition \ref{def:WG}).

In the table below, we list the families of Fano threefolds with $\rho\leq 5$ that have nontrivial Weyl group, $\wg_X$. For the remaining families, $\wg_X=0$.

\begin{table}[ht]\renewcommand{\arraystretch}{1.2}
\caption{Weyl groups of Fano threefolds} \label{weylfano}
\begin{tabularx}{0.80\textwidth}{|X|c|}
\hline
Family & $\wg_X$ \\
\hhline{|=|=|}
\textnumero 2.2, \textnumero 2.6, \textnumero 2.12, \textnumero 2.21, \textnumero 2.32, \textnumero 3.3, \textnumero 3.7, \textnumero 3.9, \textnumero 3.10, \textnumero 3.17, \textnumero 3.19, \textnumero 3.20, \textnumero 3.25, \textnumero 3.31, & $\mathbb{Z}/2\mathbb{Z}$ \\
 \textnumero 4.3, \textnumero 4.4, \textnumero 4.7, \textnumero 4.8, \textnumero 4.10, \textnumero 4.12, \textnumero 4.13, \textnumero 5.2 & \\
\hline
\textnumero 3.1, \textnumero 3.13, \textnumero 3.27, \textnumero 4.6, \textnumero 5.1 & $S_3$\\
\hline
\textnumero 4.2 & $(\mathbb{Z}/2\mathbb{Z})^2$\\
\hline
\textnumero 4.1 & $S_4$ \\
\hline
\textnumero 5.3 & $\mathbb{Z}/2\mathbb{Z} \times S_3$ \\
\hline
\end{tabularx}
\end{table}

We observe that the group $\operatorname{AutP}(X)$ is contained in Matsuki's Weyl group (cf. Proposition \ref{imagealphainweylgroup}).
The classification in Table \ref{weylfano} then implies that $\operatorname{AutP}(X)=0$ except possibly for the families listed there. For each deformation family with nontrivial $\wg_X$, we either exhibit a smooth member in the family admitting nontrivial automorphisms that induce nontrivial automorphisms of $\operatorname{Pic}(X)$, or show that any $G \subseteq \operatorname{Aut}(X)$ acts trivially on the Picard group.

\smallskip
The main classification result is as follows.

\begin{defn}\cite[III.1]{Mat95}  \label{def:WG}
    \normalfont Let $X$ be a Fano threefold. The Weyl Group of $X$, denoted as $\wg_X$, is characterized as the group of automorphisms of $\pic(Y/T)$
    \begin{enumerate}
        \item[(a)] fixing the class of the exceptional divisor for $f$,
        \item[(b)] preserving the KKMR decomposition.
    \end{enumerate}
\end{defn}

\begin{prop}\label{imagealphainweylgroup}
   Let $X$ smooth Fano threefold and $G\subseteq \aut(X)$ be a group. Then $\autp(X,G) \subseteq \wg_X$. In particular, we have $\autp(X)\subseteq \wg_X$.
\end{prop}

\begin{table}[ht]\renewcommand{\arraystretch}{1.2}
\caption{Weyl groups of Fano threefolds}  \label{weylfano}
\begin{tabularx}{0.80\textwidth}{|X|c|}
\hline
Family & $\wg_X$ \\
\hhline{|=|=|}
\textnumero 2.2, \textnumero 2.6, \textnumero 2.12, \textnumero 2.21, \textnumero 2.32, \textnumero 3.3, \textnumero 3.7, \textnumero 3.9, \textnumero 3.10, \textnumero 3.17, \textnumero 3.19, \textnumero 3.20, \textnumero 3.25, \textnumero 3.31, & $\zz/2\zz$ \\
 \textnumero 4.3, \textnumero 4.4, \textnumero 4.7, \textnumero 4.8, \textnumero 4.10, \textnumero 4.12, \textnumero 4.13, \textnumero 5.2 &  \\
\hline
\textnumero 3.1, \textnumero 3.13, \textnumero 3.27, \textnumero 4.6, \textnumero 5.1 &  $S_3$\\
\hline
\textnumero 4.2  & $(\zz/2\zz)^2$\\
\hline
\textnumero 4.1    &  $S_4$ \\
\hline
\textnumero 5.3   & $\zz/2\zz \times S_3$ \\
\hline
\end{tabularx}
\end{table}

We observe that the group $\autp(X)$ is contained in Matsuki's Weyl group (cf. Proposition \ref{imagealphainweylgroup}).
The classification in Table \ref{weylfano} then implies that $\autp(X)=0$ except possibly for the families listed there. For each deformation family with nontrivial $\wg_X$, we either exhibit a smooth member in the family admitting nontrivial automorphisms that induce nontrivial automorphisms of $\pic(X)$, or show that any $G \subseteq \aut(X)$ acts trivially on the Picard group.

In each case, we will see that $\autp(X)$ can in fact be realized by a finite subgroup of the automorphism group for which $\wg_X$ is nontrivial (see \S \ref{proof:classresult}).

\begin{remark} \label{autp:2.2exception}\begin{enumerate}[leftmargin=*]
 \normalfont\item[(i)] The result of the theorem is not true for $X$ in family \textnumero 2.2. In this case, the Kleiman-Mori cone $\neo(X)$ is generated by two extremal rays of different types, $C_1$ and $D_1$. Any automorphism of $X$ must keep each invariant, thus acts trivially on $\pic(X)$. So, $\autp(X)=0$, but $\wg_X=\zz/2\zz$.
 \item[(ii)] If $X$ is a smooth member of \textnumero 3.9 or \textnumero 4.2, we have an upper bound on $\autp(X)$ by Proposition \ref{imagealphainweylgroup}. It is not yet known whether the bound is attained in these two cases.
 \item[(iii)] A smooth Fano threefold in families \textnumero 6.1, \dots , \textnumero 10.1 is isomorphic to $\pp^1 \times S$, where $S$ is a del Pezzo surface of degree at most $5$. For all these cases, Matsuki's Weyl group coincides with the classical Weyl group of root system for the underlying $S$. It is known that the group $\autp(X)$ is usually much smaller than their complete Weyl group (see \S \ref{autP:P1timesdP}).
 \end{enumerate}
 \end{remark}

\smallskip
From Proposition \ref{imagealphainweylgroup} and Table \ref{weylfano}, we immediately get
\begin{coro} \label{im=0forsure}
 Let $X$ be a Fano threefold. Then $\autp(X)=0$ for all deformation families of Fano threefolds except possibly,
 \begin{enumerate}
 \item[(i)] $5$ families with $\rho =2$,
 \begin{center}
 \textnumero 2.2,\; \textnumero 2.6,\; \textnumero 2.12,\; \textnumero 2.21, \; \textnumero 2.32.
 \end{center}
 \item[(ii)] $12$ families with $\rho =3$,
 \begin{center}
 \textnumero 3.1,\; \textnumero 3.3,\; \textnumero 3.7,\; \textnumero 3.9,\; \textnumero 3.10,\;\textnumero 3.13,\; \textnumero 3.17,\; \textnumero 3.19,
 \; \textnumero 3.20,\; \textnumero 3.25,\; \textnumero 3.27,\; \textnumero 3.31.
 \end{center}
 \item[(iii)] $10$ families with $\rho =4$,
 \begin{center}
 \textnumero 4.1,\; \textnumero 4.2, \; \textnumero 4.3,\; \textnumero 4.4,\; \textnumero 4.6,\; \textnumero 4.7, \textnumero 4.8,\; \textnumero 4.10,\; \textnumero 4.12,\; \textnumero 4.13.
 \end{center}
 \item[(iv)] $8$ families with $\rho \geq 5$,
 \begin{center}
 \textnumero 5.1,\; \textnumero 5.2,\; \textnumero 5.3,\; \textnumero 6.1,\; \textnumero 7.1,\; \textnumero 8.1,\; \textnumero 9.1,\; \textnumero 10.1.
 \end{center}
 \end{enumerate}
\end{coro}

\section{Classification of \texorpdfstring{$\autp(X,G)$}{AutP(X,G)}} \label{proof:classresult}
\subsection {Divisors on a products of projective spaces} \label{im(alpha)divisortype}
Let $X$ be a smooth Fano threefold that is a member of any of the five deformation families
\begin{center}
\textnumero 2.6\;(a),\; \textnumero 2.32,\; \textnumero 3.3,\; \textnumero 3.17, \; \textnumero 4.1.
\end{center}
For each family, we use their Mori-Mukai description in \cite[Tables 2-4]{MM81classtable}. A smooth Fano threefold in \textnumero 2.6 is either isomorphic to a divisor of bidegree $(2,2)$ in $\pp^2\times\pp^2$ (\textnumero 2.6 (a)), or a double cover (\textnumero 2.6 (b), see Lemma \ref{autP:2-6b)}).

\begin{lemma} \label{autP:3.13}
There exists an $X$ in \textnumero 3.13 such that $\autp(X)=\wg_X \cong S_3$.
\end{lemma}
\begin{proof}
If $X$ is in family \textnumero 3.13, the center of the blow-up is a curve $C$ of bidegree $(2,2)$ and
\[
C \hookrightarrow W \hookrightarrow \mathbb{P}^2 \times \mathbb{P}^2 \xrightarrow{\operatorname{pr}_i} \mathbb{P}^2
\]
is an embedding for $i=1,2$. Recall an alternate description of $X$ from \cite[\S 5.19]{calabi}-- one can choose coordinates $(x_0:x_1:x_2],[y_0:y_1,y_2],[z_0:z_1,z_2])$ on $\pp^2 \times \pp^2 \times \pp^2$ such that $X$ can be given by the following set of equations
\[
\begin{cases}
 x_0y_0 + x_1y_1+x_2y_2=0\\
 y_0z_0+ y_1z_1 + y_2z_2 =0\\
 x_0z_1+ x_1z_0 + x_1z_2 - x_2z_1 -2x_2z_2 =0
\end{cases}
\]
with $\aut(X)\cong \mathbb{G}_a \rtimes S_3$. Here the subgroup $S_3$ is generated by the involutions
\begin{align*}
\tau_{x,y}: ([x_0:x_1:x_2],&[y_0:y_1:y_2],[z_0:z_1:z_2]) \mapsto
([y_0+2y_1+\\
&y_2:2y_0:y_0+y_2],[x_1:x_0-x_2:2x_2-x_1],[z_0:z_1:-z_2]),\\
\tau_{x,z}: .([x_0:x_1:x_2]&,[y_0:y_1:y_2],[z_0:z_1:z_2]) \mapsto \\
&([z_0:z_1:-z_2],[y_0:y_1:-y_2],[x_0:x_1:-x_2])
\end{align*}
such that its action on $\pic(\pp^2\times\pp^2\times\pp^2)$ coincides with the permutation action of $S_3$. From Proposition \ref{thm:lefschetz}, we know $\pic(X)\cong \pic(\pp^2\times\pp^2\times\pp^2)$. Thus, we have $\autp(X,G)= S_3$ where $G$ is generated by $\tau_{x,y},\tau_{x,z}$.
\end{proof}

\begin{lemma} \label{autP:4.7}
There exists an $X$ in \textnumero 4.7 such that $\autp(X)=\wg_X \cong \zz/2\zz$.
\end{lemma}
\begin{proof}
If $X$ is in family \textnumero 4.7, the center of the blow-up is a curve $C$ which is given by the disjoint union $C_1\sqcup\; C_2$ of curves $C_1$ and $C_2$ of degrees $(0,1)$ and $(1,0)$, respectively. Recall the Fano threefold $W$ from \eqref{eqn:W}, defined by $x_0y_0 + x_1y_1 + x_2y_2=0$, and the involution $\sigma \in \aut(W)$
\[
\sigma : ([x_0:x_1:x_2],[y_0:y_1:y_2]) \mapsto ([y_0:y_1:y_2],[x_0:x_1:x_2]),
\]
as in \eqref{eqn:involutionforW}.
Assume that
\begin{align*}
 C_1:= \{y_0 = y_1 =0\}, \hspace{0.5cm} C_2:= \{x_0 = x_1 =0\}.
\end{align*}
Then $\sigma$ also swaps the curves $C_1$ and $C_2$ but keeps $C_1\sqcup C_2$ invariant. Note that $C_1 \sqcup C_2$ is smooth and irreducible on $W$ since each curve $C_i$ is, and $\sigma \in \aut(W;C_1\sqcup C_2)$. Let $E_i$ denote the exceptional divisor corresponding to $C_i$, for $i=1,2$. Then $\pic(X) \cong \zz[\pi^*H_1]\oplus \zz[\pi^*H_2]\oplus \zz[E_1]\oplus \zz[E_2]$.

\subsection{Product of \texorpdfstring{$\mathbb{P}^1$}{P1} and a del Pezzo surface} \label{autP:P1timesdP}
Let $X$ be a smooth Fano threefold of the form $\mathbb{P}^1 \times S_d$, where $S_d$ is a del Pezzo surface of degree $d$, $1 \leq d \leq 9$. Then $X$ is a member of one of the following $10$ deformation families
\begin{center}
\textnumero 2.34,\; \textnumero 3.27, \; \textnumero 3.28, \; \textnumero 4.10, \; \textnumero 5.3, \; \textnumero 6.1,\; \textnumero 7.1,\; \textnumero 8.1,\; \textnumero 9.1,\; \textnumero 10.1.
\end{center}
If $X$ is in family \textnumero $2.34$, we have $d=9$ and it is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^2$. If $X$ is in family \textnumero $3.28$, $d=8$ and $X$ is isomorphic to $\mathbb{P}^1 \times \operatorname{Bl}_p\mathbb{P}^2$. We forego the discussion of these two families since each has $WG_X=0$ from \cite{Mat23}.

In each case, we will see that $\operatorname{AutP}(X)$ can in fact be realized by a finite subgroup of the automorphism group for which $\wg_X$ is nontrivial (see \S \ref{proof:classresult}).

\begin{remark}  \label{autp:2.2exception}\begin{enumerate}[leftmargin=*]
     \normalfont\item[(i)] The result of the theorem is not true for $X$ in family \textnumero 2.2. In this case, the Kleiman-Mori cone $\neo(X)$ is generated by two extremal rays of different types, $C_1$ and $D_1$. Any automorphism of $X$ must keep each invariant, thus acts trivially on $\operatorname{Pic}(X)$. So, $\operatorname{AutP}(X)=0$, but $\wg_X=\mathbb{Z}/2\mathbb{Z}$.
     \item[(ii)] If $X$ is a smooth member of \textnumero 3.9 or \textnumero 4.2, we have an upper bound on $\operatorname{AutP}(X)$ by Proposition \ref{imagealphainweylgroup}. It is not yet known whether the bound is attained in these two cases.
    \item[(iii)] A smooth Fano threefold in families \textnumero 6.1, \dots , \textnumero 10.1 is isomorphic to $\mathbb{P}^1 \times S$, where $S$ is a del Pezzo surface of degree at most $5$. For all these cases, Matsuki's Weyl group coincides with the classical Weyl group of root system for the underlying $S$. It is known that the group $\operatorname{AutP}(X)$ is usually much smaller than their complete Weyl group (see \S \ref{autP:P1timesdP}).
 \end{enumerate}
 \end{remark}

As a consequence, we can compute the first cohomology group of Picard group. We have the following result (see \S \ref{sec:H1}).
\begin{thm}  \label{thm:H1main}
    Let $X$ be a smooth Fano threefold with Picard rank $\rho \leq 5$. Let $G\subseteq \operatorname{Aut}(X)$ be a finite group. Then $H^1(G,\operatorname{Pic}(X))=0$.
\end{thm}
In fact, the group $H^1(G,\operatorname{Pic}(X))$ is known even for $\rho \geq 6$ since the first cohomology groups of del Pezzo surfaces are known.

\subsection*{Outline} We begin with some preliminaries in \S \ref{sec:preliminaries}. In \S\ref{imalphasection}, we recall the Weyl group of a smooth Fano threefold as defined by K. Matsuki. We also show that the group $\operatorname{AutP}(X)$ is bounded above by this Weyl group. Consequently, we obtain that $\operatorname{AutP}(X)$ is trivial for a majority of the families of smooth Fano threefolds. Section \ref{proof:classresult} is divided into ten subsections, each of which discusses the classification of $\operatorname{AutP}(X)$ for a class of families for which $\wg_X$ is nontrivial. In this way, we complete the proof of Theorem \ref{them:classresult}. In \S \ref{sec:H1}, we discuss the first cohomology group of Picard group of smooth Fano threefolds and provide a proof of Theorem \ref{thm:H1main}. Finally, a table summarizing the entire classification is given in \S \ref{table}.

\subsection*{Acknowledgements} I am grateful to my advisor, Alexander Duncan, for suggesting the problem and for his guidance.
I also wish to thank Kenji Matsuki for helpful correspondence regarding family \textnumero~3.13.

We ``recall the Weyl group of a smooth Fano threefold as defined by K.\ Matsuki'' and ``show that the group $\operatorname{AutP}(X)$ is bounded above by this Weyl group.'' From this, ``we obtain that $\operatorname{AutP}(X)$ is trivial for a majority of the families of smooth Fano threefolds.''

For the remaining cases where ``$\wg_X$ is nontrivial,'' Section~\ref{proof:classresult} ``is divided into ten subsections, each of which discusses the classification of $\operatorname{AutP}(X)$ for a class of families''; ``in this way, we complete the proof of Theorem~\ref{them:classresult}.''

In the exceptional families listed in Theorem~\ref{them:classresult}, the text notes: for family \textnumero~2.2, $\neo(X)$ has two extremal rays of different types and ``any automorphism of $X$ must keep each invariant, thus acts trivially on $\operatorname{Pic}(X)$,'' so $\operatorname{AutP}(X)=0$ while $\wg_X=\mathbb{Z}/2\mathbb{Z}$; for \textnumero~3.9 and \textnumero~4.2, one has ``an upper bound on $\operatorname{AutP}(X)$ by Proposition~\ref{imagealphainweylgroup}'' but it is unknown whether it is attained; and for \textnumero~6.1--\textnumero~10.1, $X\cong \mathbb{P}^1\times S$ and although Matsuki's Weyl group matches the classical Weyl group of $S$, ``$\operatorname{AutP}(X)$ is usually much smaller'' (see \S~\ref{autP:P1timesdP}).

We ``recall the Weyl group of a smooth Fano threefold as defined by K.\ Matsuki'' and ``show that the group $\operatorname{AutP}(X)$ is bounded above by this Weyl group.'' From this, ``we obtain that $\operatorname{AutP}(X)$ is trivial for a majority of the families of smooth Fano threefolds.''

For the remaining cases where ``$\wg_X$ is nontrivial,'' \section{Classification of \texorpdfstring{$\autp(X,G)$}{AutP(X,G)}}  \label{proof:classresult}
\subsection {Divisors on a products of projective spaces}   \label{im(alpha)divisortype}
Let $X$ be a smooth Fano threefold that is a member of any of the five deformation families
\begin{center}
\textnumero 2.6\;(a),\; \textnumero 2.32,\; \textnumero 3.3,\; \textnumero 3.17, \; \textnumero 4.1.
\end{center}
``is divided into ten subsections, each of which discusses the classification of $\operatorname{AutP}(X)$ for a class of families''; ``in this way, we complete the proof of the main theorem.''

In the exceptional families listed in the main theorem, the text notes: for family \textnumero~2.2, $\neo(X)$ has two extremal rays of different types and ``any automorphism of $X$ must keep each invariant, thus acts trivially on $\operatorname{Pic}(X)$,'' so $\operatorname{AutP}(X)=0$ while $\wg_X=\mathbb{Z}/2\mathbb{Z}$; for \textnumero~3.9 and \textnumero~4.2, one has ``an upper bound on $\operatorname{AutP}(X)$ by''
\begin{prop}\label{imagealphainweylgroup}
   Let $X$ smooth Fano threefold and $G\subseteq \aut(X)$ be a group. Then $\autp(X,G) \subseteq \wg_X$. In particular, we have $\autp(X)\subseteq \wg_X$.
\end{prop}
but it is unknown whether it is attained; and for \textnumero~6.1--\textnumero~10.1, $X\cong \mathbb{P}^1\times S$ and although Matsuki's Weyl group matches the classical Weyl group of $S$, ``$\operatorname{AutP}(X)$ is usually much smaller'' (see \label{autP:P1timesdP}
Let $X$ be a smooth Fano threefold of the form $\mathbb{P}^1 \times S_d$, where $S_d$ is a del Pezzo surface of degree $d$, $1 \leq d \leq 9$. Then $X$ is a member of one of the).

Question: For a Fano threefold $X$, let $\operatorname{AutP}(X)$ denote the image of the natural action homomorphism $\operatorname{Aut}(X)\to \operatorname{Aut}(\operatorname{Pic}(X))$, and let $\wg_X$ denote Matsuki's Weyl group associated to $X$. Which statement holds about deformation families of Fano threefolds?

For every deformation family of Fano threefolds other than families \textnumero 2.2, \textnumero 3.9, \textnumero 4.2, \textnumero 6.1, \textnumero 7.1, \textnumero 8.1, \textnumero 9.1, and \textnumero 10.1, there exists a member $X$ of that family such that $\operatorname{AutP}(X)=\wg_X$.

Choices:
B. For every deformation family of Fano threefolds other than families \textnumero 3.9, \textnumero 4.2, \textnumero 6.1, \textnumero 7.1, \textnumero 8.1, \textnumero 9.1, and \textnumero 10.1, there exists a member $X$ of that family such that $\operatorname{AutP}(X)=\wg_X$.
C. For every deformation family of Fano threefolds other than families \textnumero 2.2, \textnumero 3.9, \textnumero 4.2, \textnumero 6.1, \textnumero 7.1, \textnumero 8.1, \textnumero 9.1, and \textnumero 10.1, there exists a member $X$ of that family such that $\operatorname{AutP}(X)\subseteq \wg_X$.
D. For every deformation family of Fano threefolds other than families \textnumero 2.2, \textnumero 3.9, \textnumero 4.2, \textnumero 6.1, \textnumero 7.1, \textnumero 8.1, \textnumero 9.1, and \textnumero 10.1, every member $X$ of that family satisfies $\operatorname{AutP}(X)=\wg_X$.
E. For every deformation family of Fano threefolds, there exists a member $X$ of that family such that $\operatorname{AutP}(X)\subseteq \wg_X$, and equality holds except possibly for families \textnumero 2.2, \textnumero 6.1, \textnumero 7.1, \textnumero 8.1, \textnumero 9.1, and \textnumero 10.1.

dropped equality and kept only the universal inclusion into Matsuki's Weyl group

existential member-wise realization replaced by uniform statement for every member

omits the unresolved families 3.9 and 4.2 from the possible exceptions to equality

The stem gives only notation and asks which global statement is true; it does not explicitly reveal the correct exceptional-family list, equality claim, or quantifier structure.

The item is essentially a recognition task for the exact theorem statement about deformation families, with the correct option closely restating the result rather than asking for a derived conclusion from premises.

There is some pressure to compare subtle variants involving exceptions, equality vs. inclusion, and existential vs. universal quantifiers, but the task mainly tests precise recall of a classification theorem rather than genuine mathematical generation.

The distractors are plausible and well-targeted: they alter the exceptional list, weaken equality to inclusion, or change 'there exists' to 'every,' all of which reflect realistic mathematical misreadings.

A solid theorem-recall MCQ with strong distractors and no answer leakage, but it is not highly generative and is close to direct restatement of a known result.