By Leonard Roth
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Extra info for Algebraic Threefolds: With Special Regard to Problems of Rationality
One which, for every h such that 0 ~ h ~ d - 1, is variable in a linear system endowed with a well defined pure Jacobian variety J h( 5). With this connotation, let 51> 52' •.. ,5r (r > d - h) be any r general hypersurfaces on Vd ; then it may be shown that From this we may deduce equivalences for the canonical varieties of any variety Va which is in biregular (n, 1) correspondence with a variety V;. Suppose that the coincidence locus on Vd is of the form X(s -1) C~~ l ' where the numbers s may assume various values, all of them divisors of d, and where C~·~ 1 denotes an (s - I)-fold component of the coincidence locus which is non-singular and which has no inter- 1.
Systems of Surfaces. ------- As was first noticed by CASTELNUOVO (a) in the case d = 2, the properties of ~ are intimately connected with those of the PICARD varieties. These varieties will be discussed in VI: for present purposes it suffices to remark that a PICARD variety Vp is a non-singular manifold which is endowed with a completely transitive permutable continuous group of oop birational self-transformations, or automorphisms, and that the possession of such a group characterises Vp; also that Vp has a pure canonical hypersurface of order zero.
The discussion has been completed in several particulars by N OLLET . ROSENBLATT [IJ has applied analogous methods to the threefolds for which Pg ;;;; 3(q - 3), but has obtained only partial results. We have already had occasion to mention the representation of a variety Va on a multiple Wa; it will be convenient to state at this point some results, geometrical and transcendental, concerning this representation, which will be required later. Suppose then that Va is mapped on the n-fold variety ~; in general there will be a branch hypersurface on ~, possibly reducible, with components of different mUltiplicities; and, corresponding to this, a coincidence hypersurface on Va, possibly reducible, with components of different multiplicities.
Algebraic Threefolds: With Special Regard to Problems of Rationality by Leonard Roth