christian/uni
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

add rav 10

Browse Source
This commit is contained in:
christian
2022-11-24 23:00:32 +01:00
parent f5a077e53f
commit f6e36c24b0
9 changed files with 437 additions and 1 deletions
Download Patch File Download Diff File Expand all files Collapse all files
ws2022/rav/lecture/rav.pdf
BIN
View File
Binary file not shown.
ws2022/rav/lecture/rav.tex
+1
View File
@@ -24,5 +24,6 @@ Christian Merten (\href{mailto:cmerten@mathi.uni-heidelberg.de}{cmerten@mathi.un
\input{rav7.tex}
\input{rav8.tex}
\input{rav9.tex}
\input{rav10.tex}
\end{document}
ws2022/rav/lecture/rav10.pdf
BIN
View File
Binary file not shown.
ws2022/rav/lecture/rav10.tex
+378
View File
@@ -0,0 +1,378 @@
\documentclass{lecture}
\begin{document}
\section{Examples of algebraic varieties}
\begin{aufgabe}[]
Let $f\colon X \to Y$ be a morphism of algebraic pre-varieties. Assume
\begin{enumerate}[(i)]
\item $Y$ is a variety.
\item There exists an open covering $(Y_i)_{i \in I}$ of $Y$ such that the open subset
$f^{-1}(Y_i)$ is a variety.
\end{enumerate}
Show that $X$ is a variety.
\end{aufgabe}
\begin{aufgabe}[]
Let $X$ be a topological space. Assume that there exists a covering $(X_i)_{i \in I}$ of
$X$ by irreducible open subsets such that for all $(i,j)$, $(X_i \cap X_j) \neq \emptyset$.
Show that $X$ is irreducible.
\end{aufgabe}
\subsection{Grassmann varieties}
Let $0 \le p \le n$ be integers. The Grassmannian $\text{Gr}(p, n)$ is the set
of $p$-dimensional linear subspaces of $k^{n}$. In order to endow this set with a structure
of algebraic prevariety, there are various possibilities:
\begin{enumerate}[(i)]
\item To a $p$-dimensional linear subspace $E \subseteq k^{n}$, we associate the line
$\Lambda^{p} E \subseteq \Lambda^{p} k^{n} \simeq k^{\binom{n}{p}}$, which
defines a point in the projective space $k\mathbb{P}^{\binom{n}{p}-1}$.
Claim: The map $\text{Gr}(p, n) \to k\mathbb{P}^{\binom{n}{p} -1}$
is an injective map whose image is a Zariski-closed subset of $k\mathbb{P}^{\binom{n}{p} -1}$.
This identifies $\text{Gr}(n, p)$ canonically to a projective variety. In particular
one obtains in this way a structure of \emph{algebraic variety}
on $\text{Gr}(p, n)$.
\item For the second approach, recall that $\text{GL}(n, k)$ acts transitively on
$\text{Gr}(p, n)$. But the identification of $k^{n}$ to $(k^{n})^{*}$
via the canonical basis of $k^{n}$ enables one to define, for all $E \in \text{Gr}(p, n)$,
a canonical complement $E^{\perp} \in \text{Gr}(n-p, n)$, i.e.
an $(n-p)$-dimensional linear subspace such that $E \oplus E^{\perp} = k^{n}$.
So the stabiliser of $E \in \text{Gr}(p, n)$ for the action of
$\text{GL}(n, k)$ is conjugate to the subgroup
\begin{salign*}
\text{P}(p, n) \coloneqq
\left\{ g \in \text{GL}(n, k) \middle \vert
\begin{array}{l}
g = \begin{pmatrix} A & B \\ 0 & C \end{pmatrix} \\
\text{with } A \in \text{GL}(p, k), B \in \text{Mat}(p \times (n-p), k),\\
\text{and } C \in \text{GL}(n-p, k)
\end{array}
\right\}
.\end{salign*}
This shows that the Grassmannian $\text{Gr}(p, n)$ is a homogeneous space
under $\text{GL}(n, k)$ and that
\begin{salign*}
\text{Gr}(p, n) \simeq \text{GL}(n, k) / \text{P}(p, n)
\end{salign*}
which is useful if one knows that, given an affine algebraic group $G$ and
a closed subgroup $H$, the homogeneous space $G / H$ is an algebraic variety. We
will come back to this later on.
\item The third uses the gluing theorem. In particular, it also constructs
a standard atlas on $\text{Gr}(p, n)$, like the one we had on
$k\mathbb{P}^{n-1} = \text{Gr}(1, n)$.
The idea is that, in order to determine a $p$-dimensional subspace of $k^{n}$,
it suffices to give a basis of that subspace, which is a family of $p$ vectors
in $k^{n}$. Geometrically, this means that the subspace in question is seen
as the graph of a linear map $A\colon k^{p} \to k^{n}$.
Take $E \in \text{Gr}(p, n)$ and let $(v_1, \ldots, v_p)$ be a basis of $E$ over $k$.
Let $M$ be the $(n \times p)$-matrix representing the coordinates
of $(v_1, \ldots, v_p)$ in the canonical basis of $k^{n}$. Since $M$ has rank $p$,
there exists a $(p \times p)$-submatrix of $M$ with non-zero determinant: We set
\begin{salign*}
J &\coloneqq \{ \text{indices } j_1 < \ldots < j_p \text{ of the rows of that submatrix}\} \\
M_J &\coloneqq \text{the submatrix in question}
.\end{salign*}
Note that if $M' \in \text{Mat}(n \times p, k)$ corresponds to a basis
$(v_1', \ldots, v_p')$, there exists a matrix $g \in \text{GL}(p, k)$ such that
$M' = Mg$. But then $(M')_J = (Mg)_J = M_J g$, so
\[
\text{det }(M')_J = \text{det } (M_J g) = \text{det}(M_J) \text{det}(g)
,\]
which is non-zero if and only if $\text{det}(M_J)$ is non-zero. As a consequence,
given a subset $J \subseteq \{1, \ldots, n\} $ of cardinal $p$, there is a well-defined
subset
\begin{salign*}
G_J \coloneqq \left\{ E \in \text{G}(p, n) \mid
\exists M \in \text{Mat}(n \times p, k), E = \text{im }M \text{ and }
\text{det}(M_J) \neq 0
\right\}
.\end{salign*}
Moreover, if $M$ satisfies the conditions $E = \text{im }M$ and
$\text{det}(M_J) \neq 0$, then
$(M M_J^{-1})_J = I_p$ and $\text{im}(MM^{-1}_J) = \text{im }M = E$.
In fact, if $E \in G_J$, there is a unique matrix $N \in \text{Mat}(n \times p, k)$,
such that $E = \text{im }N$ and $N_J = I_p$, for if $N_1, N_2$ are two
such matrices, the columns of $N_2$ are linear combinations of those of $N_1$,
thus $\exists g \in \text{GL}(p, k)$ such that $N_2 = N_1g$. But then
\[
I_p = (N_2)_J = (N_1g)_J = (N_1)_J g = g
.\]
So, there is a well-defined map
\begin{salign*}
\hat{\varphi}_J: G_J &\longrightarrow \operatorname{Hom}(k^{J}, k^{n}) \\
E &\longmapsto N \text{ such that } E = \text{im }N \text{ and } N_J = I_p
\end{salign*}
whose image can be identified to the subspace
$\text{Hom}(k^{J}, k^{J^{c}})$, where $J^{c}$ is the complement of $J$ in
$\{1, \ldots, n\} $, via the map $N \mapsto N_{J^{c}}$. Conversely, a
linear map $A \in \text{Hom}(k^{J}, k^{J^{c}})$ determines a rank $p$ map
$N \in \text{Hom}(k^{J}, k^{n})$ such that $N_J = I_p$ via the formula
$N(x) = x + Ax$.
Geometrically, this means that the $p$-dimensional subspace
$\text{im }N \subseteq k^{n}$ is equal to the graph of $A$.
This also means that we can think of $G_J$ as the set
\begin{salign*}
\{E \in \text{Gr}(p, n) \mid E \cap k^{J^{c}} = \{0_{k^{n}}\} \}
.\end{salign*}
The point is that $\text{im } \hat{\varphi}_J = \text{Hom}(k^{J}, k^{J^{c}})$
can be canonically identified with the affine space $k^{p(n-p)}$ and that we
have a bijection
\begin{salign*}
\varphi_J \colon G_J &\xlongrightarrow{\simeq} \text{Hom}(k^{J}, k^{J^{c}})
\simeq k^{p(n-p)} \\
E &\longmapsto A \mid \text{gr}(A) = E \\
\text{gr}(A) &\longmapsfrom A
.\end{salign*}
Note that the matrix $N \in \text{Mat}(n \times p, k)$
such that $\text{im }N = E$ and $N_J = I_p$
is row-equivalent to $\begin{pmatrix} I_p \\ A \end{pmatrix} $
with $A \in \text{Mat}((n-p) \times p, k)$.
Now, if $E \in G_{J_1} \cap G_{J_2}$, then, for all
$M \in \text{Mat}(p \times n, k)$ such that $\text{im } M = E$,
$\hat{\varphi}_{J_1}(E) = M M_{J_1}^{-1}$ and
$\hat{\varphi}_{J_2}(E) = M M_{J_2}^{-1}$. So
\begin{salign*}
\text{im } \hat{\varphi}_{J_1}
&= \left\{ N \in \text{Hom}(k^{J_1}, k^{n}) \mid N_{J_1} = I_p,
\text{im } N_{J_1} = E \text{ and }
\text{det}(N_{J_2}) \neq 0
\right\} \\
&= \{ N \in \text{im } \hat{\varphi}_{J_1} \mid \text{det}(N_{J_2}) \neq 0\}
\end{salign*}
which is open in $\text{im } \hat{\varphi}_{J_1} \simeq \text{im } \varphi_{J_1}$.
Moreover, for all $N \in \text{im }\hat{\varphi}_{J_1}$,
\[
\hat{\varphi}_{J_2} \circ \hat{\varphi}_{J_1}^{-1}(N) = N N_{J_2}^{-1}
\] and, by Cramer's formulae, this is a regular function
on $\text{im }\hat{\varphi}_{J_1}$.
We have therefore constructed a covering
\[
\text{Gr}(p, n) = \bigcup_{J \subseteq \{1, \ldots, n\}, \# J = p } G_J
\]
of the Grassmannian $\text{Gr}(p, n)$ by subsets $G_J$
that can be identified to the affine variety $k^{p(n-p)}$ via bijective
maps $\varphi_J\colon G_j \to k^{p(n-p)}$ such that,
for all $(J_1, J_2)$, $\varphi_{J_1}(G_{J_1} \cap G_{J_2})$ is open
in $k^{p(n-p)}$ and the map
$\varphi_{J_2} \circ \varphi_{J_1}^{-1}\colon \varphi_{J_1}(G_{J_1} \cap G_{J_2}) \to \varphi_{J_2}(G_{J_1} \cap G_{J_2})$
is a morphism of affine varieties. By the gluing theorem,
this endows $\text{Gr}(p, n)$ with a structure of algebraic prevariety.
\end{enumerate}
\subsection{Vector bundles}
\begin{definition}[]
A \emph{vector bundle} is a triple
$(E, X, \pi)$ consisting of two algebraic varieties $E$ and $X$, and
a morphism $\pi\colon E \to X$ such that
\begin{enumerate}[(i)]
\item for $x \in X$, $\pi^{-1}(\{x\} )$ is a $k$-vector space.
\item for $x \in X$, there exists an open neighbourhood $U$ of $x$
and an isomorphism of algebraic varieties
\[
\Phi\colon \pi^{-1}(U) \xlongrightarrow{\simeq} U \times \pi^{-1}(\{x\} )
\] such that
\begin{enumerate}[(a)]
\item $\text{pr}_1 \circ \Phi = \pi |_{\pi^{-1}(U)}$ and
\item for $y \in U$, $\Phi|_{\pi^{-1}(\{y\})}\colon \pi^{-1}(\{y\})
\to \{y\} \times \pi^{-1}(\{x\})$ is
an isomorphism of $k$-vector spaces.
\end{enumerate}
\end{enumerate}
A morphism of vector bundles is a morphism of algebraic varieties $f\colon E_1 \to E_2$
such that $\pi_2 \circ f = \pi_1$ and $f$ is $k$-linear in the fibres.
\end{definition}
\begin{bem}
In practice, one often proves that a variety $E$ is a vector bundle over $X$ by
finding a morphism $\pi\colon E \to X$ and an open covering
\[
X = \bigcup_{i \in I} U_i
\] such that $E|_{U_i} \coloneqq \pi^{-1}(U_i)$ is isomorphic to
$U_i \times k^{n_i}$ for some integer $n_i$, in such a way that, on $U_i \cap U_j$,
the morphism
\[
\Phi_j \circ \Phi_i^{-1}\Big|_{\Phi_i(\pi^{-1}(U_i \cap U_j))}\colon
(U_i \cap U_j) \times k^{n_i} \longrightarrow
(U_i \cap U_j) \times k^{n_j}
\] is an isomorphism of algebraic varieties such that the following diagram commutes
and $\Phi_j \circ \Phi_i^{-1}$ is linear fibrewise:
\[
\begin{tikzcd}
(U_i \cap U_j) \times k^{n_i} \arrow{dr}{\text{pr}_1} \arrow{rr}{\Phi_j \circ \Phi_i^{-1}}
& & (U_i \cap U_j) \times k^{n_j} \arrow{dl}{\text{pr}_1}\\
& U_i \cap U_j & \\
\end{tikzcd}
.\] In particular $k^{n_i} \simeq k^{n_j}$ as $k$-vector spaces, so
$n_i = n_j$ if $U_i \cap U_j \neq \emptyset$, and
$\Phi_j \circ \Phi_i^{-1}$ is necessarily of the form
\[
(x, v) \longmapsto (x, g_{ji}(x) \cdot v)
\] for some morphism of algebraic varieties
\[
g_{ji}\colon U_i \cap U_j \longrightarrow \text{GL}(n, k)
.\]
These maps $(g_{ij})_{(i, j) \in I \times I}$ then
satisfy for $x \in U_i \cap U_j \cap U_l$
\[
g_{lj}(x) g_{ji}(x) = g_{li}(x)
\] and for $x \in U_i$, $g_{ii}(x) = \text{I}_n$.
\end{bem}
\begin{satz}
If $\pi\colon E \to X$ is a morphism of algebraic varieties and
$X$ has an open covering $(U_i)_{i \in I}$ over which $E$ admits
local trivialisations
\[
\Phi_i \colon E|_{U_i} = \pi^{-1}(U_i) \xlongrightarrow{\simeq} U_i \times k^{n}
\]
with $\text{pr}_1 \circ \Phi_i = \pi|_{\pi^{-1}(U_i)}$
such that the isomorphisms
\[
\Phi_j \circ \Phi_i^{-1} \colon (U_i \cap U_j) \times k^{n}
\longrightarrow (U_i \cap U_j) \times k^{n}
\] are
linear in the fibres, then for all $x \in X$, $\pi^{-1}(\{x\})$ has
a well-defined structure of $k$-vector space and the local trivialisations
$(\Phi_i)_{i \in I}$ are linear in the fibres. In particular,
$E$ is a vector bundle.
\end{satz}
\begin{proof}
For $x \in U_i$ and $a, b \in \pi^{-1}(\{x\})$, let
\[
a + \lambda b \coloneqq \Phi_i^{-1}(x, \text{pr}_2 (\Phi_i(a)) + \lambda \text{pr}_2 (\Phi_i(b)))
.\]
By using the linearity in the fibres of $\Phi_j \circ \Phi_i^{-1}$, one verifies
that this does not depend on the choice of $i \in I$.
\end{proof}
\begin{bem}[]
Assume given an algebraic prevariety $X$ obtained by gluing affine varieties
$(X_i)_{i \in I}$ along isomorphisms $\varphi_{ji}\colon X_{ij} \xrightarrow{\simeq} X_{ji}$
defined on open subsets $X_{ij} \subseteq X_i$,
such that $X_{ii} = X_i$, $\varphi_{ii} = \text{Id}_{X_i}$
%, $\varphi_{ji}(X_{ij})$ is open in $X_{ji}$
and
$\varphi_{lj} \circ \varphi_{ji} = \varphi_{li}$ on $X_{ij} \cap X_{il} \subseteq X_i$.
Recall that such an $X$ comes equipped with a canonical
map $p \colon \bigsqcup_{i \in I} \to X$ such that
$p_i \coloneqq p|_{X_i}\colon X_i \to X$ is an isomorphism onto an affine open subset
$U_i \coloneqq p_i(X_i) \subseteq X$ and, if we set $\varphi_i = p_i^{-1}$,
we have $\varphi_j \circ \varphi_i^{-1} = \varphi_{ji}$
on $\varphi_i(U_i \cap U_j)$.
Let us now consider the vector bundle $X_i \times k^{n}$ on each of the affine varieties
$X_i$ and assume that an isomorphism of algebraic prevarieties of the form
\begin{salign*}
\Phi_{ji}\colon X_{ij} \times k^{n} &\longrightarrow X_{ji} \times k^{n} \\
(x, v) &\longmapsto (\varphi_{ji}(x), h_{ji}(x) \cdot v)
\end{salign*}
has been given, where $h_{ij}\colon X_{ij} \to \text{GL}(n, k)$
is a morphism of algebraic varieties, in such a way that the following compatibility
conditions are satisfied:
\begin{salign*}
\Phi_{ii} = \text{Id}_{X_{ii} \times k^{n}}
\end{salign*}
and, for all $(i, j, l)$ and all $(x, v) \in (X_{ij} \cap X_{il}) \times k^{n}$
\[
\Phi_{lj} \circ \Phi_{ji}(x, v) = \Phi_{li}(x, v)
.\]
Then there is associated to this gluing data an algebraic vector bundle
$\pi\colon E \to X$, endowed with
local trivialisations $\Phi_i \colon E|_{U_i} \xrightarrow{\simeq} U_i \times k^{n}$,
where as earlier $U_i = p(X_i) \subseteq X$,
in such a way that, for all $(i, j)$ and all $(\xi, v) \in (U_i \cap U_j) \times k^{n}$,
\[
\Phi_j \circ \Phi_i^{-1}(\xi, v) =
(\xi, g_{ji}(\xi) \cdot v)
\] where $g_{ji}(x) = h_{ji}(\varphi_i(\xi)) \in \text{GL}(n, k)$, so
$g_{ii} = \text{I}_n$ on $U_i$, and, for all $(i, j, l)$ and
all $\xi \in U_i \cap U_j \cap U_l$,
\begin{salign*}
g_{lj}(\xi) g_{ji}(\xi) &= h_{lj}(\varphi_j(\xi)) h_{ji}(\varphi_i(\xi)) \\
&= h_{lj}(\varphi_{ji}(\varphi_i(\xi))) h_{ji}(\varphi_i(\xi)) \\
&= h_{li}(\varphi_i(\xi)) \\
&= g_{li}(\xi)
.\end{salign*}
Indeed, we can simply set
\begin{salign*}
E \coloneqq \left( \bigsqcup_{i \in I} X_i \times k^{n} \right) / \sim
\end{salign*}
where $(x, v) \sim (\varphi_{ji}(x), h_{ji}(x) \cdot v)$, and, by the
gluing theorem, this defines an algebraic prevariety, equipped
with a morphism $\pi\colon E \to X$ induced
by the first projection $\text{pr}_1\colon \bigsqcup_{i \in I} (X_i \times k^{n})
\to \bigsqcup_{i \in I} X_i$.
The canonical map $\hat{p}\colon \bigsqcup_{ i \in I} (X_i \times k^{n}) \to E$
makes the following diagram commute
\[
\begin{tikzcd}
\bigsqcup_{i \in I} (X_i \times k^{n}) \arrow{d}{\text{pr}_1}
\arrow{r}{\hat{p}} & E \arrow{d}{\pi} \\
\bigsqcup_{i \in I} X_i \arrow{r}{p} & X \\
\end{tikzcd}
\]
and it induces an isomorphism of prevarieties
\[
\hat{p}|_{X_i \times k^{n}}\colon X_i \times k^{n}
\xrightarrow{\simeq} E|_{p(X_i)}
= \pi^{-1}(p(X_i))
\]
such that $\pi \circ \hat{p}|_{X_i \times k^{n}} = p|_{X_i} \circ \text{pr}_1$.
Since $p|_{X_i}$ is an isomorphism between $X_i$ and the open subset
$U_i = p(X_i) \subseteq X$ with inverse $\varphi_i$, the
isomorphism $\hat{p}|_{X_i \times k^{n}}$
induces a local trivialisation
\begin{salign*}
\Phi_i \colon E|_{U_i} &\longrightarrow U_i \times k^{n} \\
w &\longmapsto (\pi(w), v)
\end{salign*}
where $v$ is defined as above by $\hat{p}(x, v) = w$. Note that $p(x) = \pi(w)$ in this
case, and that $\pi^{-1}(\{\pi(w)\}) \simeq k^{n}$
via $\Phi|_{\pi^{-1}(\{\pi(w)\})}$. As the isomorphism of algebraic prevarieties
\[
\Phi_j \circ \Phi_i^{-1}\colon (U_i \cap U_j) \times k^{n}
\longrightarrow (U_i \cap U_j) \times k^{n}
\]
thus defined is clearly linear fibrewise, we have indeed constructed in this way
a vector bundle $\pi\colon E \to X$, at least in the category of algebraic prevarieties.
Note that if the prevariety $X$ obtained via the gluing of the $X_i$ is
a variety, then we can show that $E$ is actually a variety
(because the product variety $U_i \times k^{n}$ is separated). The rest of the verifications,
in particular the fact that for all $(\xi, v) \in U_i \cap U_j \times k^{n}$
\[
\Phi_j \circ \Phi_i^{-1}(\xi, v) = (\xi, h_{ji}(\varphi_i(\xi)) \cdot v)
\] is left to the reader.
\end{bem}
\begin{aufgabe}[]
Consider the set
\[
E \coloneqq \{ (\rho, v) \in k \mathbb{P}^{1} \times k\mathbb{P}^{2} \mid v \in \rho\}
\] and the canonical map $\pi\colon E \to k\mathbb{P}^{1}$.
Show that $E$ is a vector bundle on $k\mathbb{P}^{1}$ and compute
its ,,cocycle of transition functions`` $g_{10}$ on the standard atlas
$(U_0, U_1)$ of $k\mathbb{P}^{1}$ with
\begin{salign*}
\varphi_{10}\colon k \setminus \{0\} &\longrightarrow k \setminus \{0\} \\
t &\longmapsto \frac{1}{t}
.\end{salign*}
\end{aufgabe}
\end{document}