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