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Abelian affine open cover affine open subset affine scheme ample assume birational morphism canonical closed immersion closed point closed subscheme closed subset closed subvariety closure coherent 0x-module complete variety coordinate system Corollary to Theorem curve CVmodule D+(a defined Definition denoted dimension effective divisor element exact sequence Example exists fc[X fibered variety finite number finite type Furthermore Hence homogeneous hypothesis integral domain invertible sheaf irreducible components isomorphism Lemma Let f linear system maximal ideal nonsingular variety nonzero normal variety obtains open cover open dense subset open neighborhood open subset phism polynomial ring presheaf prime divisor prime ideal Proj projective Proof proper birational Proposition prove quadric transformations quasi-coherent Rat(K rational map ring homomorphism ringed space S-scheme satisfies sheaves singular points strictly rational map subring Suppose surjective surjective morphism topology