Algebraic Threefolds: With Special Regard to Problems of by Leonard Roth

By Leonard Roth

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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.

We pass now to the form of the RIEMANN-RoCH theorem for more general systems of surfaces; this is due to B. SEGRE [2J, the proof being on the lines of CASTELNUOVO'S demonstration of the analogous result in the theory of surfaces. An essential preliminary is the theorem (1,6) that the deficiency of the characteristic system of any complete linear system (at least ( 0 2) of irreducible surfaces cannot exceed q2; this is proved by projecting V generically into a primal W, and showing that adjoint primals of sufficiently high order cut on W a linear system the defect of whose characteristic system is precisely q2' and then applying the following lemma: 32 III.

In point of fact, SEVERI'S relation is most easily established by using the above equivalence for 1], together with TODD'S formuTa for P,. (ROTH [14J). If the latter formula is to be avoided, an alternative definition of P,. is required, except when the number can be found from first principles, as in an example worked by B. SEGRE [1J. B. SEGRE proves that, if V is transformed birationally into a nonsingular threefold V' in such a way as to introduce a single exceptional surface of either species, then [1] J = ['YJ' J, i.

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