By Leonard Roth
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Additional info for Algebraic Threefolds: With Special Regard to Problems of Rationality
The standard forms of NOETHER and ENRIQUES. 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.
Must terminate after a finite number of stages. It may be shown (ENRIQUES*) [2J) that the last surface of the series has rational, elliptic or hyperelliptic curve sections, or is representable on a double plane with a branch curve which is either a non-singular plane quartic or a sextic with two consecutive triple points (i. e. a double quadric cone with sextic of genus 4 as branch curve). These results may be summarised as follows: *) Actually the result is due to this paper. CASTELNUOVO; it is given in an appendix to IV.
It lies on Bs; or it is a contact of this curve with each of the curves of if>1> if>j passing through P; and conversely. In this last case these two curves can coincide with ~, and then P is double for the g(~~l cut by if> on ~; otherwise P lies on 9\; and conversely. A. A. + (9\5) . A. varies on the Jacobian curve ~ of the net cut by (9 on 51' Denoting by T, T* the surfaces described by st, st*, we therefore have (TT*) == 9\ Again the Jacobian curve ~ is given by the equivalence ~== 3~ Substituting for ~ and + 58 + ~ .
Algebraic Threefolds: With Special Regard to Problems of Rationality by Leonard Roth