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    The $d$-uple embedding is an isomorphism onto its image

    P-AGH34DUPLEISO

    Area
    Algebraic Geometry
    Topics
    Isomorphism, Morphisms, Veronese Embedding
    Status
    draft

    Show that the \(d\)-uple embedding of \(\PP^n\) is an isomorphism onto its image.

    Solution

    Let \[ \rho_d:\PP^n\longrightarrow\PP^N \] be the \(d\)-uple embedding from P-AGH212DUPLE, and let \(Y=\rho_d(\PP^n)\). For \(0\le i,j\le n\), write \(y_{ij}\) for the homogeneous coordinate of \(\PP^N\) corresponding to the monomial \(x_i^{d-1}x_j\); in particular \(y_{ii}\) corresponds to \(x_i^d\).

    1. The open subsets\[ U_i=Y\cap D_+(y_{ii}) \]cover \(Y\).

    Every point of \(Y\) has the form \(\rho_d([a_0:\cdots:a_n])\). Some \(a_i\) is nonzero, and then its \(y_{ii}\)-coordinate is \(a_i^d\ne0\). Thus the point lies in \(U_i\).

    2. On \(U_i\), the formula\[ \psi_i([y])=[y_{i0}:\cdots:y_{in}] \]defines a morphism \(\psi_i:U_i\to\PP^n\).

    On \(U_i\) the coordinate \(y_{ii}\) is nonzero, so the displayed homogeneous coordinates are not all zero. On the affine target chart where the \(i\)th coordinate is nonzero, the coordinate ratios of \(\psi_i\) are \[ \frac{y_{ij}}{y_{ii}}, \] which are regular functions on \(D_+(y_{ii})\). Hence \(\psi_i\) is a morphism.

    3. The maps \(\psi_i\) are the restrictions of one inverse morphism \(\psi:Y\to\PP^n\).

    Let \([a]=[a_0:\cdots:a_n]\) with \(a_i\ne0\). For \(y=\rho_d([a])\), the defining coordinates satisfy \[ y_{ij}=a_i^{d-1}a_j, \] and therefore \[ \psi_i(y) =[a_i^{d-1}a_0:\cdots:a_i^{d-1}a_n] =[a_0:\cdots:a_n]. \] Thus each \(\psi_i\) is the set-theoretic inverse of \(\rho_d\) on \(U_i\). In particular the \(\psi_i\) agree on overlaps \(U_i\cap U_j\), so they glue to a morphism \(\psi:Y\to\PP^n\). The same computation gives \(\psi\circ\rho_d=\operatorname{id}_{\PP^n}\) and \(\rho_d\circ\psi=\operatorname{id}_Y\).

    4. Q.E.D.

    Step 3 exhibits a morphism inverse to \(\rho_d\), so the \(d\)-uple embedding is an isomorphism onto its image.

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