By Leonard Roth

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**Extra resources for Algebraic Threefolds: With Special Regard to Problems of Rationality**

**Sample text**

A general series of equivalence is then defined as an aggregate of virtual sets obtained by addition and subtraction of a finite number of elementary series. Series (of points) and systems (of curves) of equivalence on a threefold are defined in an analogous manner. The theory of equivalence on a threefold V which, in all that follows, is assumed to be non-singular, requires for its development a knowledge of the corresponding results for curves and surfaces; in particular, the establishment of the invariant series and systems of V rests on the theory of invariants of curves and surfaces.

Chapter III. Systems of Surfaces. 1. 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.

For example, B. SEGRE [IJ uses the present methods to obtain equivalences for the invariant series of any surface of the form c1 Sl + C2 S2 ' where Cv C2 are integers, in terms of the invariant series of Sl and S2' In the same work SEGRE finds the covariant systems of one or two nets of surfaces on V, of two or more pencils, and also of linear systems of freedom three or four, thereby establishing many interesting relations between the entities in question. One of the most striking of these is the following: given two pencils IAI, IBI of general character, generically situated, the number of pairs A, B which have stationary contact with one another is 48(P,.