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By Igor V. Dolgachev

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3. Q-DIVISORS 41 The multiplication maps OX (D)⊗OX (D ) → OX (D +D ) define a graded sheaf of OX -algebras A(X, D) = OX (iD). 16) i∈Z Passing to global sections we can define the graded algebra ∞ A(X, D) = L(iD). 17) i=0 It is important to understand that this is a direct sum, although all the graded parts are subspaces in the field of rational functions K on X. One may view A(X, D) as a subalgebra of the field K(T ) by considering an isomorphic graded subalgebra of K(T ) ∞ L(iD)T i . e. there exists an open cover (Ui )i∈I of X such that the image of D under (1) (1) the restriction map ZX → ZUi , i ∈ I, is linearly equivalent to zero.

3. A closed integral subscheme X of P = Pnk is called projectively normal if the restriction homomorphism Γ∗ (OP (1)) ∼ = k[T0 , . . , Tn ] → Γ∗ (OX (1)) is surjective. Let IX be the sheaf of ideals defining X. The ideal IX = Γ∗ (IX ) in k[T0 , . . , Tn ] is called the homogeneous ideal of X and the quotient ideal k[X] = k[T0 , . . , Tn ]/IX is called the homogeneous coordinate algebra of X. The Corollary implies that X is projectively normal if and only if k[X] is normal. One checks that the fields of fractions of k[X] and Γ∗ (OX (1)) are both isomorphic to k(X)(t), where k(X) is the field of rational functions on X.

Then OX (en) are invertible sheaves for all n and the multiplication maps OX (en)⊗OX (em) → OX ((n+m)e) are isomorphisms. Let OX (1)⊗e → OX (e) be the multiplication map. Passing to the double duals we get a map OX (1)[e] → OX (e). A nonzero map OX (D) → OX (D ) exists if and only if D − D is effective. Thus De − eD1 = d s Es for some positive integers ds . Similarily, considering the maps OX (1)⊗i → OX (i) for all i > 0 and passing to the double duals, we obtain Di − iD1 ≥ 0. The maps OX (i)⊗e → OX (ie) = OX (iDe ) show that Die − eDi ≥ 0.