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100644 --- a/motives.tex +++ b/motives.tex @@ -64,6 +64,7 @@ \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Trd}{Trd} \DeclareMathOperator{\sing}{sing} +\DeclareMathOperator{\Bl}{Bl} %\DeclareFontFamily{OT1}{pzc}{} %\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.2] pzcmi7t}{} @@ -2938,7 +2939,7 @@ R_{3,3}(\{a,b,c\})\{i\}. Для квадрики Пфистера верно аналогичное замечание. \end{remark} -\begin{theorem}[Зайнуллин--Петров--Семенов] +\begin{theorem}[Зайнуллин--Петров--Семенов]\label{thm:ZPS} Пусть $G$~--- полупростая алгебраическая группа над $F$, $X$~--- $G$-однородное проективное многообразие, клеточное над общей точкой, $p$~--- простое число. @@ -3253,5 +3254,526 @@ P(R_2({}_{\xi}G), t) = \frac{1-t^{2\cdot 15}}{1-t^{15}} = 1 + t^{15}, Получается некоторый инвариант в $H^5(F,\mathbb{Z}/2)$. \end{example} +% 16.04.2012 + +\subsection{Набросок доказательства теоремы~\ref{thm:ZPS}} + +Во-первых, нам понадобится <<теорема Крулля--Шмидта>>. +О какой категории идет речь? +Зафиксируем ${}_{\xi}G$ и рассмотрим категорию мотивов +по модулю простого числа $p$, а в ней~--- псевдоабелеву +подкатегорию, порожденную мотивами ${}_{\xi}G$-однородных +многообразий. +То есть, мы берем замыкание относительно взятия прямых слагаемых и +свдигов (и тензорных произведений, хотя это не важно). + +\begin{theorem}[Черноусов--Меркурьев]\label{thm:ChM} +В этой категории выполнена теорема Крулля--Шмидта: +если есть разложение объекта в прямую сумму неразложимых, +то оно единственно. +\end{theorem} +Пусть теперь $X,Y$~--- проективные однородные многообразия, +$X\xrightarrow{f} Y$~--- локально (по Зарискому) тривиальное +расслоение с клеточным слоем $F$. +В силу клеточности $F$, его мотив раскладывается в +сумму сдвигов мотивов Тейта: +\[ +M(F) = \bigoplus\mathbb{Z}\{i\}. +\] +Тогда $M(X) = M(Y)\otimes M(F) = \bigoplus M(Y)\{i\}$. +Поэтому, в частности, +$\CH(X) = \CH(Y)\otimes\CH(F)$ как $\CH(Y)$-модуль. +Это можно понять явно: +если наше расслоение тривиализуется над $U\subseteq Y$, +то морфизм $U\times F \to U$ дает нам возможность +по элементу $a\in\CH^*(F)$ построить +цикл $1\times a$ и рассмотреть его замыкание в $X$. +Такие замыкания как раз и образуют $\CH(Y)$-базис в $\CH(X)$, +если в качестве $a$ брать элементы базиса $\CH(F)$. + +Пусть теперь $X$~--- ${}_{\xi}G$-однородное проективное +многообразие: $X = {}_{\xi}(G/P)$. +Рассмотрим расслоение +${}_{\xi}(G/b) \to X$ со слоем $P/B$. +Над $F(X)$ наш торсор становится тривиальным, +поэтому расслоение выглядит как $G/B \to G/P$. +Оно локально тривиально по Зарискому, поэтому +это верно и над некоторым открытым $U\subseteq X$. +Теперь мы можем подействовать группой $G$ и покрыть все $X$ +такими открытыми. +Значит, все отображение локально тривиально по Зарискому. +По теореме~\ref{thm:ChM} $M({}_{\xi}(G/B)) = M(X)\otimes M(P/B)$. +Поэтому, если $M({}_{\xi}(G/B))\otimes(\mathbb{Z}/p)$ +раскладывается в сумму сдвигов $R_p(\xi)$, то и $M(X)$ +тоже (по теореме Крулля--Шмидта). + +Значит, достаточно для случая $X = {}_{\xi}(G/B)$ доказать, +что $M(X)\otimes(\mathbb{Z}/p)$ раскладывается в сумму +неразложимых кусков $R_p(\xi)$ со сдвигами. +Рассмотрим коммутативную диаграмму +\[ +\begin{tikzcd} +\CH^*(BB)/p \arrow{r} \arrow[equal]{d} +& \CH^*({}_{\xi}(G/B))/p \arrow[->>]{r} \arrow{d} +& \CH^*(\xi)/p \arrow{d} \\ +\CH^*(BB)/p \arrow{r} +& \CH^*(G/B)/p \arrow[->>]{r} +& \CH^*(G)/p. +\end{tikzcd} +\] +\begin{enumerate} +\item +Мы знаем, что над замыканием +$\CH^*(G)/p = (\mathbb{Z}/p)[x_1,\dots,x_r] / (x_i^{p^{k_i}})$, +и образ $\CH^*(BB)/p$ в $\CH^*(G/B)/p$ +состоит из рациональных циклов. +Обозначим через $e_i$ любой прообраз элемента $x_i$. +\item Из определения $J_p(\xi) = (j_1,\dots,j_r)$ мы знаем, +что $e_i^{p^{j_i}} + \mbox{добавка}$~--- тоже рациональный цикл. +У $\CH^*(G/B)/p$ как $\CH^*(BB)/p$-модуля есть система образующих +(точнее, базис над образом $\CH^*(BB)/p$), состоящая +из элементов вида $e^I=e_1^{j_1}\dots e_r^{i_r}$, +где $I = (i_1,\dots,i_r)$, и $0\leq I \leq p^K-1$ +(то есть, $0\leq i_m \leq p^{k_m}-1$ для всех $m$). +\item Ищем рациональные циклы на $X\times X$ при помощи +метода общей точки: +\[ +\begin{tikzcd} +\CH^*(X\times X) \arrow[->>]{r} \arrow{d}{\res} +& \CH^*(X_{F(X)}) \arrow{d}{\isom} \\ +\CH^*(G/B\times G/B) \arrow[->>]{r} +& \CH^*((G/B)_{F(X)}). +\end{tikzcd} +\] +Возьмем какой-нибудь прообраз $\alpha_i\in\CH^*(X\times X)$ +элемента $e_i\in\CH^*((G/B)_{F(X)})$, +и пусть $\ol{\alpha_i} = \res(\alpha_i)$~--- его образ +в $\CH^*(G/B\times G/B)$. +Тогда $\ol{\alpha_i} = e_i \times 1 + \mbox{добавка}$. +Более аккуратные рассуждения показывают, что +$\ol{\alpha_i} = e_i \times 1 + (\mbox{добавка}) - 1\times e_i$. +\item Пусть цикл $a\in\im(\CH^*(BB)/p)$ рационален. +Можно найти <<двойственный>> к нему +цикл $\alpha^{\vee} \in\im(\CH^*(BB)/p)$ такой, что +$\deg(e^{p^k-1}\cdot a \cdot a^{\vee}) = 1$. +То есть, на $\im(\CH^*(BB)/p)$ есть невырожденная билинейная +форма со значениями в $\mathbb Z/p$. +Тогда цикл +\[ +q = \alpha^{p^J-1} \cdot +\left(e^{p^JM}a\times e^{p^J(p^{K-J}-1-M)}a^\vee\right) +\] +рационален, поскольку $\alpha^{p^J-1}$ рационален, +и степени $e^{p^J}$ рациональны (по определению +$J$-инварианта). +Нетрудно проверить, что $q$~--- проектор +(если проигнорировать добавки). +Построение $q$ зависит от произвольных $M$ и $a$ из образа. +Если брать все возможные $M$ такие, что $0\leq M \leq p^{K-J}-1$, +и $a$ из базиса образа, получается +набор попарно ортогональных проекторов, которые в сумме +дают диагональ. + +Упражнение: некоторая степень любого элемента конечного моноида +является идемпотентом +(на самом деле, нужно брать не такую степень, а еще большую +степень этого, чтобы убить добавки). + +Посчитаем $q\cdot q$. +Можно считать, что $\alpha = e\times 1 - 1\times e$. +Тогда +$\alpha^{p^J-1} = \sum_{I} e^I \times e^{p^J-1-I}$, +и поэтому +\[ +q = \sum_I e^{p^J M + I} a +\times e^{p^K-p^J M - 1 - I}a^{\vee} += \deg(e^{p^K-1} a a^{\vee})(\dots). +\] +\end{enumerate} + +\subsection{Группы типа $\mathsf{F}_4$} + +Приведем пример многообразия, не расщепимого над общей точкой. +Будем обозначать расщепимые октонионы через $\mathbb{O}$, +а их компактную форму (над $\mathbb{R}$) через $C$. +Напомним, что над $\mathbb{R}$ +бывают такие группы типа $\mathsf{F}_4$: +\begin{enumerate} +\item $H_3(\mbox{компактная форма},C)$: +$\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\draw (1)--(2); +\draw (3)--(4); +\draw (2) edge[transform canvas={yshift=2pt}] (3); +\draw (2) edge[transform canvas={yshift=-2pt}] (3); +\draw ($\sm*(2.0, 0.2)$) -- ($\sm*(2.2, 0)$) -- ($\sm*(2.0, -0.2)$); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture}$ + +\item $H_3(\mathbb{O})$: +$\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\draw (1)--(2); +\draw (3)--(4); +\draw (2) edge[transform canvas={yshift=2pt}] (3); +\draw (2) edge[transform canvas={yshift=-2pt}] (3); +\draw ($\sm*(2.0, 0.2)$) -- ($\sm*(2.2, 0)$) -- ($\sm*(2.0, -0.2)$); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture}$ + +\item $H_3(\mbox{гиперболическая форма}, C)$: +$\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\draw (1)--(2); +\draw (3)--(4); +\draw (2) edge[transform canvas={yshift=2pt}] (3); +\draw (2) edge[transform canvas={yshift=-2pt}] (3); +\draw ($\sm*(2.0, 0.2)$) -- ($\sm*(2.2, 0)$) -- ($\sm*(2.0, -0.2)$); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\draw [black] (4) circle (5.0pt); +\end{tikzpicture} +$ +\end{enumerate} +Что нарисовано в последнем случае? +Существует поле $F$ (например, $F=\mathbb{R}$) и группа типа +$\mathsf{F}_4$ над ним такие, что параболическая подгруппа +типа $P_4$ определена над $\mathbb{R}$, а остальные~--- нет. +Пусть $\xi\in H^1(F,\mathsf{F}_4)$~--- коцикл, +$X = {}_{\xi}(\mathsf{F}_4/P_4)$. +Тогда $\xi_{F(X)}$ может дать группу с индексом +$\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\draw (1)--(2); +\draw (3)--(4); +\draw (2) edge[transform canvas={yshift=2pt}] (3); +\draw (2) edge[transform canvas={yshift=-2pt}] (3); +\draw ($\sm*(2.0, 0.2)$) -- ($\sm*(2.2, 0)$) -- ($\sm*(2.0, -0.2)$); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\draw [black] (4) circle (5.0pt); +\end{tikzpicture}$. +По модулю $2$ есть два инварианта: +\begin{itemize} +\item $f_3\colon H^1(F, \mathsf{F}_4) \to H^3(F, \mathbb{Z}/2)$; +\item $f_5\colon H^1(F, \mathsf{F}_4) \to H^5(F, \mathbb{Z}/2)$. +\end{itemize} +Если у поля нет расширений нечетной степени, то это чистые формы. +Поэтому есть $3$-кратная и $5$-кратная формы Пфистера, +ассоциированные с $\xi$. + +Пусть $J$~--- 27-мерная йорданова алгебра. +Предполагаем, что она приведенная (reduced). +Наш коцикл $\xi\in H^1(F,\mathsf{F}_4)$ +удовлетворяет следующим равносильным условиям: +\begin{enumerate} +\item +Каноническое отображение $H^1(F,\mathsf{F}_4) \to H^1(F,\mathsf{E}_6)$ +переводит наш коцикл $\xi$ +в изотропную группу с индексом Титса +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\draw (1)--(3)--(4)--(5)--(6); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\draw [black] (1) circle (5.0pt); +\draw [black] (6) circle (5.0pt); +\end{tikzpicture} +\] +\item $g_3(\xi)=0$, где +$g_3\colon H^1(F,\mathsf{F}_4) \to H^3(F,\mu_3^{\otimes 2})$. +Этого всегда можно добиться кубическим расширением; +а если нет расширений нечетной степени, то это заведомо так. +\item Пусть $Q_x(y) = xyx$~--- квадратичная операция в $J$. +Элемент $e\in J$ называется \term{идемпотентом ранга $1$}, +если $Q_e(J)\leq Fe$. Это равносильно тому, что +$N(e)=0$ и $e^{\sharp}=0$ (или тому, что $N(e,e,x)=0$ для всех +$x\in J$). +Рассмотрим многообразие +\[ +\{\la e\ra \mid e\mbox{ --- идемпотент ранга $1$}\} = +{}_{\xi}(\mathsf{E}_6/P_6). +\] +Условие состоит в том, что у него есть рациональная точка. +\end{enumerate} +Поэтому внутри ${}_{\xi}(\mathsf{E}_6/P_6)$ есть $\mathbb{G}_m$, +которая дает разложение вида +$J = Fe\oplus U\oplus V$ по весовым пространствам $\mathbb{G}_m$. +При этом $\dim U = 16$, $\dim V = 10$. +Будем записывать элементы $J$ как тройки +вида $(\alpha,b,c)$ в соответствии с этим разложением. +Заметим, что ${}_{\xi}(\mathsf{E}_6/P_6)$~--- замкнутое +подмногообразие в $\mathbb{P}(J)$. +Следующая фильтрация будет $\Spin_{10}$-инвариантной: +\[ +\begin{tikzcd}[column sep=0.4em] +{}_{\xi}(\mathsf{E}_6/P_6) +& \arrow{d}{\mathbb{A}^{16}} +& \{\alpha=0\} \arrow[left hook->]{ll} +& \arrow{d}{\mathbb{A}^{5}} +& \{\alpha=0,b=0\} = \Spin_{10}/P_1\arrow[left hook->]{ll} \\ +& \{b=0,c=0\} = \pt +& +& \{\alpha=0,c=0\} = \Spin_{10}/P_5 +\end{tikzcd} +\] +Можно начать с другого конца и получить такую фильтрацию: +\[ +\begin{tikzcd}[column sep=0.4em] +{}_{\xi}(\mathsf{E}_6/P_6) +& \arrow{d}\mathbb{A}^{8} +& \{c=0\} \arrow[left hook->]{ll} +& \arrow{d}\mathbb{A}^{1} +& \{c=0,b=0\} = \pt \arrow[left hook->]{ll} \\ +& \{b=0,\alpha=0\} = \Spin_{10}/P_1 +& +& \{c=0,\alpha=0\} = \Spin_{10}/P_5 +\end{tikzcd} +\] +Кстати, $\Spin_{10}/P_1$~--- это восьмимерная пфистерова квадрика, +которая и дает отображение $f_3$. + +Если у ${}_{\xi}(\mathsf{E}_6/P_6)$ есть рациональная точка, +то есть большая клетка, и оно бирационально эквивалентно +проективному пространству: +\begin{eqnarray*} +{}_{\xi}(\mathsf{E}_6/P_6) & \dashrightarrow +& \mathbb{P}(Fe\oplus U),\\ +{}[\alpha:b:c] & \mapsto & [\alpha:b],\\ +{}[\alpha^2:\alpha b:Q_b(e)] & \mapsfrom & [\alpha:b]. +\end{eqnarray*} +Локус стрелки $[\alpha:b:c]\mapsto [\alpha:b]$~--- восьмимерная +квадрика. +Локус стрелки $[\alpha:b]\mapsto [\alpha^2:\alpha b:Q_b(e)]$~--- +это $\{\alpha=0, Q_b(e)=0\} = \Spin_{10}/P_5$~--- скрученная +форма максимального ортогонального грассманиана. + +Многообразие ${}_{\xi}(\mathsf{F}_4/P_4)$ живет внутри +${}_{\xi}(\mathsf{E}_6/P_6)$ как гиперплоское сечение +вида $\alpha + t(c) = 0$, где $t$~--- линейная форма на $V$ +(напомним, что $\dim V = 10$). +Давайте пересечем все с этим уравнением. +Получим бирациональное отображение +\[ +{}_{\xi}(\mathsf{F}_4/P_4) \dashrightarrow Q. +\] +Локус слева~--- семимерная квадрика $\Spin_9/P_1$. +Локус справа~--- пятнадцатимерная квадрика +$\{\alpha^2 + t(Q_b(e)) = 0\} = \Spin_{10}/P_5 = \Spin_9 / P_4$. +Здесь $Q$~--- норменная квадрика: +мы начали с формы Пфистера $\lAngle a,b,c,d,e\rAngle$. +Семнадцатимерная +квадратичная форма $\la a\ra\perp \lAngle b,c,d,e\rAngle$~--- +инвариант этой формы. +Ассоциированная с ней пятнадцатимерная квадрика и есть $Q$. + +Что дальше? +Оказывается, $\Bl_{\Spin_9/P_1}({}_{\xi}(\mathsf{F}_4/P_4)) \isom +\Bl_{\Spin_9/P_4}(Q)$. +Для мотива раздутия есть формула вида +\[ +M(\Bl_X(Y)) = M(Y) \oplus ( M(X)\otimes M(\mathbb{P}(N_XY))\{\dots\}). +\] +Мотив норменной квадрики раскладывается в один экземпляр +мотива Роста $R(\lAngle a,b,c,d,e\rAngle)$ (он как раз +пятнадцатимерный) и $R(\lAngle b,c,d,e\rAngle)$ +в нужном количестве (7 штук): +\[ +\begin{tikzpicture}[scale=1.0,thin,font=\scriptsize] +\def\sm{0.7} +\coordinate (0) at ($\sm*(0, 0)$); +\coordinate (1) at ($\sm*(1.4, 0)$); +\coordinate (2) at ($\sm*(2.8, 0)$); +\coordinate (3) at ($\sm*(4.2, 0)$); +\coordinate (4) at ($\sm*(5.6, 0)$); +\coordinate (5) at ($\sm*(7.0, 0)$); +\coordinate (6) at ($\sm*(8.4, 0)$); +\coordinate (7) at ($\sm*(9.8, 0)$); +\coordinate (8) at ($\sm*(11.2, 0)$); +\coordinate (9) at ($\sm*(12.6, 0)$); +\coordinate (10) at ($\sm*(14.0, 0)$); +\coordinate (11) at ($\sm*(15.4, 0)$); +\coordinate (12) at ($\sm*(16.8, 0)$); +\coordinate (13) at ($\sm*(18.2, 0)$); +\coordinate (14) at ($\sm*(19.6, 0)$); +\coordinate (15) at ($\sm*(21.0, 0)$); +\coordinate (p1) at ($\sm*(6.8, 1)$); +\coordinate (p2) at ($\sm*(7.0, 1)$); +\coordinate (p3) at ($\sm*(7.2, 1)$); +\coordinate (p4) at ($\sm*(15.2, 1)$); +\coordinate (p5) at ($\sm*(15.4, 1)$); +\coordinate (p6) at ($\sm*(15.6, 1)$); +\draw (0)--(1)--(2)--(3)--(4)--(5)--(6)--(7)--(8)--(9)--(10)--(11)--(12)--(13)--(14)--(15); +\draw[dashed] (0) edge[bend left=45](15); +\draw[dotted] (1) edge[bend left=35](8); +\draw[dotted] (2) edge[bend left=35](9); +\draw[dotted] (7) edge[bend left=35](14); +\node at (0) [below=5pt] {$0$}; +\node at (1) [below=5pt] {$1$}; +\node at (2) [below=5pt] {$2$}; +\node at (3) [below=5pt] {$3$}; +\node at (4) [below=5pt] {$4$}; +\node at (5) [below=5pt] {$5$}; +\node at (6) [below=5pt] {$6$}; +\node at (7) [below=5pt] {$7$}; +\node at (8) [below=5pt] {$8$}; +\node at (9) [below=5pt] {$9$}; +\node at (10) [below=5pt] {$10$}; +\node at (11) [below=5pt] {$11$}; +\node at (12) [below=5pt] {$12$}; +\node at (13) [below=5pt] {$13$}; +\node at (14) [below=5pt] {$14$}; +\node at (15) [below=5pt] {$15$}; +\foreach \point in +{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{p1,p2,p3,p4,p5,p6} + { + \fill [black] (\point) circle (0.7pt); + } +\end{tikzpicture} +\] +Мотив норменной квадрики $\Spin_9/P_1$ раскладывается +в $R(\lAngle b,c,d,e\rAngle)$ (1 штука) и мотивы Роста, +соответствующие $f_3$. +Многообразие $\Spin_9/P_4$ расщепимо над общей точкой. +Мотив ${}_{\xi}(\mathsf{F}_4/P_4)$ выглядит так: +\[ +\begin{tikzpicture}[scale=1.0,thin,font=\scriptsize] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\coordinate (5) at ($\sm*(5.2, 1)$); +\coordinate (6) at ($\sm*(6.6, 1)$); +\coordinate (7) at ($\sm*(8.0, 1)$); +\coordinate (8) at ($\sm*(9.4, 1)$); +\coordinate (9) at ($\sm*(10.8, 1)$); +\coordinate (10) at ($\sm*(12.2, 1)$); +\coordinate (11) at ($\sm*(13.6, 1)$); +\coordinate (12) at ($\sm*(15.0, 1)$); +\coordinate (5x) at ($\sm*(5.2, -1)$); +\coordinate (6x) at ($\sm*(6.6, -1)$); +\coordinate (7x) at ($\sm*(8.0, -1)$); +\coordinate (8x) at ($\sm*(9.4, -1)$); +\coordinate (9x) at ($\sm*(10.8, -1)$); +\coordinate (10x) at ($\sm*(12.2, -1)$); +\coordinate (11x) at ($\sm*(13.6, -1)$); +\coordinate (12x) at ($\sm*(15.0, -1)$); +\coordinate (13) at ($\sm*(16.0, 0)$); +\coordinate (14) at ($\sm*(17.4, 0)$); +\coordinate (15) at ($\sm*(18.8, 0)$); +\coordinate (16) at ($\sm*(20.2, 0)$); +\node at ($\sm*(0, -2)$) {$0$}; +\node at ($\sm*(1.4, -2)$) {$1$}; +\node at ($\sm*(2.8, -2)$) {$2$}; +\node at ($\sm*(4.2, -2)$) {$3$}; +\node at ($\sm*(5.2, -2)$) {$4$}; +\node at ($\sm*(6.6, -2)$) {$5$}; +\node at ($\sm*(8.0, -2)$) {$6$}; +\node at ($\sm*(9.4, -2)$) {$7$}; +\node at ($\sm*(10.8, -2)$) {$8$}; +\node at ($\sm*(12.2, -2)$) {$9$}; +\node at ($\sm*(13.6, -2)$) {$10$}; +\node at ($\sm*(15.0, -2)$) {$11$}; +\node at ($\sm*(16.0, -2)$) {$12$}; +\node at ($\sm*(17.4, -2)$) {$13$}; +\node at ($\sm*(18.8, -2)$) {$14$}; +\node at ($\sm*(20.2, -2)$) {$15$}; +\draw (1)--(2)--(3)--(4)--(5)--(6)--(7)--(8)--(9)--(10)--(11)--(12)--(13)--(14)--(15)--(16); +\draw (4)--(5x)--(6x)--(7x)--(8x)--(9x)--(10x)--(11x)--(12x)--(13); +\draw[dashed] (1) edge[bend left=45](16); +\draw[dotted] (2) edge[bend left=45](5); +\draw[dotted] (3) edge[bend left=45](6); +\draw[dotted] (4) edge[bend right=35](7); +\draw[dotted] (8) edge[bend left=45](11); +\draw[dotted] (9) edge[bend left=45](12); +\draw[dotted] (10) edge[bend right=35](13); +\draw[dotted] (5x) edge[bend right=45](8x); +\draw[dotted] (6x) edge[bend right=45](9x); +\draw[dotted] (7x) edge[bend left=45](10x); +\draw[dotted] (11x) edge[bend right=45](14); +\draw[dotted] (12x) edge[bend right=45](15); +\foreach \point in +{1,2,3,4,5,6,7,8,9,10,11,12,5x,6x,7x,8x,9x,10x,11x,12x,13,14,15,16} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture} +\] +Предположим, что у ${}_{\xi}(\mathsf{F}_4/P_4)$ есть точка +над $F'$, где $F'/F$~--- расширение нечетной степени. +Верно ли, что у него есть точка над $F$? +Если есть два проективных гладких многообразия $Y_1,Y_2$, +которые бирационально эквивалентны, то +$Y_1(F)\neq\varnothing$ тогда и только тогда, +когда $Y_2(F)\neq\varnothing$. +Теперь из ${}_{\xi}(\mathsf{F}_4/P_4)(F')\neq\varnothing$ +следует, что $Q(F')\neq\varnothing$, +и по теореме Спрингера $Q(F)\neq\varnothing$, +откуда ${}_{\xi}(\mathsf{F}_4/P_4)(F)\neq\varnothing$. + +Мы получили картинку +$$ +\begin{tikzcd} +\Bl_{\Spin_9/P_1}({}_{\xi}(\mathsf{F}_4/P_4)) \arrow{r}{\isom}\arrow{d} +& \Bl_{\Spin_9/P_4}Q\arrow{d} \\ +{}_{\xi}(\mathsf{F}_4/P_4) & Q +\end{tikzcd} +$$ + \end{document}