By Iwaniec H., Kowalski E.

This ebook exhibits the scope of analytic quantity concept either in classical and moderb path. There are not any department kines, in reality our cause is to illustrate, partic ularly for rookies, the interesting numerous interrelations.

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Finally, we observe that a variety Vd which is unirational and representable on an involution In" in an extension K', may be represent- V Va. Va *) In general, however. only in some extension of K: see B. SEGRE [J OJ. 2. The standard forms of N OETHER and E NRIQUES. 41 able on an involution In'" where n" < n', in some other extension K"; in particular, we may have n" = 1, in which case Va is birational in K". For example, a non-singular cubic surface V~ of S3 is representable on an involution 16 in K(P), and is representable on an involution 12 in K(l) (§ 3), while it projects birationally on to a plane by means of transversals to a pair of skew lines of V~.

With regard to the surfaces of the third and fourth families, all that we need note at present is that those of the third family contain a rational pencil of conics, determinable in K, while those of the fourth family contain a rational pencil of elliptic curves, likewise determinable in K. 3. Varieties which are unirational in K(Sh). In the previous section we have encountered various examples of surfaces which are either unirational or birational in the extension K' = K(P); this is a particular case of an extension which has proved to be of considerable importance in the theory, for it frequently happens that a given variety Vd is unirational or birational in K(Sh) where Sh (h~ 0) is the generic member of a system of spaces 5 h lying on Vd.

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.

### Analytic number theory by Iwaniec H., Kowalski E.

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