By Leonard Roth

ISBN-10: 3540018891

ISBN-13: 9783540018896

ISBN-10: 3642855318

ISBN-13: 9783642855313

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

**Example text**

The fact that the non-singular cubic primal V~ (d ~ 3) is unirational was known to NOETHER; a far more remarkable result, due to MORIN [7J, states that the general primal of any given order is unirational provided it lies in a space of sufficiently high dimension. The first step towards this theorem was taken in MORIN [1J, where it is shown that - in a convenient extension K' of K - the general quartic primal of 5 r is unirational, and representable on 16' provided that r ~ 7, i. e. as soon as the primal contains planes; again, in MORIN [2J, it is shown that the general quintic primal of 5 r is unirational provided that r~ 17, i.

43 VII. n = 8: there are two species, the first mapped by cubics (P(O), the second by quartics

If, further, v = 1, and 11;, is birational in K(Sh)' then Vd is birational. As a first application we consider the non-singular cubic primal V~ (d~ 2), for which we introduce the extension K(P), where P is the general point of V~. The general tangent plane to V~ at P meets Vj in an irreducible cubic with a node at P; and this is birational in K(P). Again, the tangent primes to vj at points of the cubic cut Vj in a birational system, of index 6, of monoids, each of which is birational in K(P). It follows that Vj is unirational in K(P) and, more precisely, representable in K(P) upon an involution 16 • *) In ENRIQUES [I J it is stated that this surface is birational in K; the above modification is given in ENRIQUES [7J.

### Algebraic Threefolds: With Special Regard to Problems of Rationality by Leonard Roth

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