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    The complement of a hypersurface in $\PP^n$ is affine

    P-AGH35COMPLHYP

    Area
    Algebraic Geometry
    Topics
    Affine Varieties, Hypersurfaces, Veronese Embedding
    Status
    draft

    By abuse of language, a variety is affine if it is isomorphic to an affine variety. If \(H \subseteq \PP^n\) is any hypersurface, show that \(\PP^n \sm H\) is affine.

    Hint

    Let \(H\) have degree \(d\), consider the \(d\)-uple embedding of \(\PP^n\) in \(\PP^N\), and use that \(\PP^N\) minus a hyperplane is affine.

    Solution

    Let \(H=Z(f)\), where \(f\in k[x_0,\ldots,x_n]\) is a nonzero homogeneous polynomial of degree \(d\). Let \[ \rho_d:\PP^n\longrightarrow\PP^N \] be the \(d\)-uple embedding, and write \(Y=\rho_d(\PP^n)\).

    1. There is a hyperplane \(K\subseteq\PP^N\) such that\[ \rho_d(H)=Y\cap K. \]

    Index the homogeneous coordinates \(y_0,\ldots,y_N\) of \(\PP^N\) by the degree-\(d\) monomials \(M_0,\ldots,M_N\) in \(x_0,\ldots,x_n\). Since these monomials form a basis of the degree-\(d\) homogeneous polynomials, write \[ f=\sum_{i=0}^N c_iM_i. \] Let \(L=\sum_i c_i y_i\) and put \(K=Z(L)\). For every \(P\in\PP^n\), \[ L(\rho_d(P))=\sum_i c_iM_i(P)=f(P). \] Hence \(P\in H\) exactly when \(\rho_d(P)\in K\), which proves the displayed identity.

    2. The complement \(Y\setminus(Y\cap K)\) is an affine variety.

    The image \(Y\) is closed in \(\PP^N\) by P-AGH212DUPLE. A projective linear change of coordinates carries \(K\) to a coordinate hyperplane, whose complement is the standard affine chart \(\AA^N\). Therefore \(\PP^N\setminus K\) is affine.

    Now \[ Y\setminus(Y\cap K)=Y\cap(\PP^N\setminus K). \] Because \(Y\) is closed in \(\PP^N\), this intersection is closed in the affine variety \(\PP^N\setminus K\). A closed subvariety of an affine variety is affine, so the displayed complement is affine.

    3. Q.E.D.

    By P-AGH34DUPLEISO, \(\rho_d\) is an isomorphism \(\PP^n\cong Y\). Step 1 shows that it restricts to an isomorphism \[ \PP^n\setminus H \cong Y\setminus(Y\cap K), \] and step 2 shows that the target is affine. Hence \(\PP^n\setminus H\) is affine.

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    • P-AGH37HYPMEETSA positive-dimensional projective variety meets every hypersurfaceuses