By Leonard Roth

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**Extra info for Algebraic Threefolds: With Special Regard to Problems of Rationality**

**Example text**

We shall show, however, in § 3 that the surface is unirational in K(P) and representable on an involution 16 , III. n = 4: from a point P of itself the surface projects into a cubic surface, on which we are given a line p, image of P; hence, by § 3, the DEL PEZZO quartic surface is representable in K(P) on an involution 1 2 , IV. n = 5: the DEL PEZZO quintic possesses a single apparent triple point, which means that we can determine, in K, a pair of points P, Q of the surface. Projecting the surface, in its normal space 55' from P, Q, we obtain a cubic surface on which we are given a pair of skew lines p, q, images of P, Q; and this surface may be mapped birationally on a plane by drawing transversals to p and q.

S"J as follows: when h = 0, the variety is simply the Jacobian set of 150 1, while, when h > 0, it consists of the points P such that the h + 1 tangent [d - 1Js to the members of the pencils which pass through P have a common [d - h]. It may then be shown that the (possibly virtual) variety X" defined by X" == 1,,[50 ,51>'" ,S"J - K"l~(1 + sy -1], varies in a system of equivalence {X,,} which is independent of the pencils ISJ It may further be shown that the systems {X,,} possess the property of adjunction expressed by the relation (on 5).

Covariant point sets. well known formula for the genus of a reducible curve. We then extend the definition to the case where