By Norman Earl Steenrod

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In Section 3, we prove the theorem of Segre. Essentially the same argument, with minor modifications to be made afterwards, proves the following stronger theorem of Manin (1966). 1 For an arbitrary normal projective variety the Picard number is defined as the rank of the N´eron–Severi group, the group of Cartier divisors up to numerical equivalence. For varieties with h 1 (X, O X ) = 0 the two definitions coincide. 2. Two smooth cubic surfaces defined over a perfect field, each of Picard number one, are birationally equivalent if and only if they are projectively equivalent.

For arbitrary X , the spaces (X, ⊗m X ) are usually hard to compute because the ⊗m X have quite high rank. Therefore it is important to have similar criteria which involve line bundles only. The natural candidate is the canonical bundle ω X = ∧n X of highest degree K¨ahler differential forms, which is always defined over the fixed ground field. For smooth X , the canonical bundle is represented by a divisor K X defined over the given ground field, and it is convenient to denote it by O X (K X ).

Prove that X is rational. 7 Numerical criteria for nonrationality Rationality and unirationality force strong numerical constraints on a variety. Let X = X/k be the sheaf of regular differential forms (K¨ahler differentials) on a variety X over k. 52. If a smooth projective variety X is rational, then it has no nontrivial global K¨ahler one-forms. In fact, the space of global sections ⊗m (X, ⊗m X ) of the sheaf X is zero for all m ≥ 1. The same holds for unirational X , provided the ground field has characteristic zero.