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    Linear, algebraic, and numerical equivalence

    D-DIVEQUIV

    Let \(X\) be a smooth projective variety over an algebraically closed field, with intersection pairing \(D \cdot C = \deg \restrictionof{\OO_X(D)}{C}\) for \(C\) an irreducible curve. Let \(D_1,D_2\) be divisors on \(X\).

    • \(D_1\) and \(D_2\) are linearly equivalent, \(D_1 \sim D_2\), when \(D_1 - D_2 = \div(f)\) for a rational function \(f\).

    • \(D_1\) and \(D_2\) are algebraically equivalent, \(D_1 \sim_{\mathrm{alg}} D_2\), when they are related by the equivalence relation generated by the following: there are a nonsingular connected curve \(T\), points \(t_1,t_2\in T\), and a divisor \(\mathcal D\) on \(X\times T\), flat over \(T\), whose fibres over \(t_1\) and \(t_2\) are \(D_1\) and \(D_2\).

    • \(D_1\) and \(D_2\) are numerically equivalent, \(D_1 \equiv D_2\), when \((D_1 - D_2) \cdot C = 0\) for every irreducible curve \(C \subseteq X\).

    The Néron–Severi group is \(\NS(X) = \Div(X)/{\sim_{\mathrm{alg}}} = \Pic(X)/\Pic^0(X)\), a finitely generated abelian group by the theorem of the base; its rank is the Picard number \(\rho(X)\). A divisor \(D\) is nef when \(D \cdot C \geq 0\) for every irreducible curve \(C\), and big when \(\limsup_{m\to\infty} h^0(\OO(mD))/m^{\dim X}>0\).

    Linear equivalence implies algebraic equivalence, which implies numerical equivalence. The group \(\Pic(X)\) of linear equivalence classes can have a positive-dimensional component \(\Pic^0(X)\); the quotient \(\NS(X)\) is finitely generated, and the group of numerical classes is \(\NS(X)\) modulo its torsion subgroup.

    Ampleness and nefness depend only on the numerical class: by the Nakai–Moishezon criterion, \(D\) is ample if and only if \(D^{\dim V}\cdot V>0\) for every closed subvariety \(V\) of positive dimension. In \(\NS(X)\otimes\RR\), the nef classes form the closure of the cone of ample classes (Kleiman), and a divisor is big if and only if some positive multiple is linearly equivalent to an ample divisor plus an effective divisor (Kodaira’s lemma).

    On a smooth projective curve, algebraic and numerical equivalence both mean equality of degree, so \(\NS(X)=\ZZ\) by degree; linear equivalence is finer when the genus is positive, and nef means \(\deg\ge0\).

    Define linear, algebraic, and numerical equivalence of divisors.
    What is the Neron--Severi group?
    What is a nef divisor? A big divisor?

    Dependencies

    • D-DIVAMPLEAmple and very ample divisorsuses
    • D-5PQ5WWeil and Cartier divisors, and when they agreeuses

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