By Leonard Roth

ISBN-10: 3540018891

ISBN-13: 9783540018896

ISBN-10: 3642855318

ISBN-13: 9783642855313

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LEFSCHETZ also proves that a non-singular surface which is a complete intersection of r - 2 primals, and which is normal in Sr (r > 3) in general contains only complete intersections, i. e. provided the surface is a generic member of the linear system determined by it on a threefold containing it. LEFSCHETZ obtains an analogous result for surfaces of order exceeding 3 in ordinary space, thereby making precise an incomplete theorem due to N OETHER. 4. Theory of the base. The existence of a finite base for algebraic curves on an algebraic surface was first established by SEVERI, using transcendental methods; later a topological proof, originating with LEFSCHETZ and simplified by ALBANESE, was obtained (for the details see ALBANESE [4J).

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. e. for any such transformation P,. is an absolute invariant. 4. Further developments. The six invariant entities lXI, ISXI, ml, In 11]1, I" defined above form a complete set in the sense that a wide class of enumerative and functional problems can be solved in terms of these alone. 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.

The RIEMANN-RoCH theorem. 31 sections with any other component; then it may be proved that Xh(V';z) == X h(Vl) + 1:(s - 1) Xh(C~~ 1) , where X h (vl) denotes the transform of X h (Vl). This result (RoTH [36J) generalises a relation which, in the case h = d - 1, is classical. 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.

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

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