By Leonard Roth
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Additional resources for Algebraic Threefolds: With Special Regard to Problems of Rationality
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~.
An essential first step in the development of criteria of rationality for three folds and higher varieties is the birational transformation, in K, of a rational curve or surface to one or other of a certain number of standard forms. First, let V~ be a non-singular rational curve in higher space, so that its general projection W~ on to a plane will be a curve of order n, with ~ (n -1) (n - 2) ordinary nodes. The adjoint curves of order n - 2 cut on W~ a linear series g~=~, determinable in K, which, if n > 3, can be mapped on the prime sections of a non-singular normal curve V~-2.
The locus of points at which a surface of (9 and a surface of (91 have a common tangent line. It is clear that SR passes simply through (X and (Xl' while the Jacobian curves 58' 5 81 of (9 and (91 do not contain any of these points. Through any point P not belonging to (X or (Xl there passes a single characteristic curveQl of (9 and a single characteristic curveQllof (91' If P lies on 58 or 5", Ql orQll (as the case may be) has a double point at P, and conversely; the points P of SR which do not also lie on 58 or 5 81 are characterised by the fact that the corresponding curves Ql, Qll are tangent there; and a point P which lies on SR and 5.
Algebraic Threefolds: With Special Regard to Problems of Rationality by Leonard Roth