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The RIEMANN-RoCH theorem. We consider m this section the problem of determining the freedom of the complete linear system characterised by a given non-singular surface on a non-singular threefold V. If C is such a surface, with virtual characters n, n, p, we define the virtual freedom d of the system ICI by the formula d = n - n + p - Pa + 2, where Pa denotes the arithmetic genus of V. In the case where C is non-special, with effective freedom r = d, we say that ICI is regular. It is known that there exist regular systems on V; thus (SEVERI [IJ) the surfaces adjoint to any sufficiently high multiple of a linear system which is free from fundamental surfaces form a regular system.

To begin with, it has long been known (see BAKER, a) that non-singular quadrics of any dimension exceeding 2, and also quadrics with only a certain degree of specialisation, contain only complete intersections. Next, it has been shown by SEVERI [4J that any non-singular threefold of S4 contains only complete intersections, from which it follows that an analogous property holds for a non-singular primal of any dimension. SEVERI [10J has extended this result to primals of any dimension r ;;; 4 which contain fewer than 00' - 3 multiple points.

Thus, let 5 be a general hypersurface of Vd i. e. 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.

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