\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}