Algebraic Geometry: Proceedings of the Conference at Berlin by F. A. Bogomolov, A. N. Landia (auth.), H. Kurke, J. H. M.

By F. A. Bogomolov, A. N. Landia (auth.), H. Kurke, J. H. M. Steenbrink (eds.)

The convention on Algebraic Geometry, held in Berlin 9-15 March 1988, used to be organised through the Sektion Mathematik of the Humboldt-Universitat. The establishing committee consisted of H. Kurke, W. Kleinert, G. Pfister and M. Roczen. The convention is one in a sequence organised by means of the Humboldt-Universitat at ordinary periods of 2 or 3 years, with the aim of offering a gathering position for mathematicians from jap and western international locations. the current quantity comprises embellishments of a part of the lectures awarded on the convention and a few articles on comparable matters. All papers have been topic to the typical refereeing technique of Compositio Mathematica, and H. Kurke acted as a visitor editor of this magazine. The papers specialise in real subject matters in algebraic geometry and singularity idea, similar to vector bundles, arithmetical algebraic geometry, intersection thought, moduli and Hodge concept. we're thankful to all those that, by means of their hospitality, their presence on the Con­ ference, their help or their written contributions, have made this convention to successful. The editors Compositio Mathematica seventy six: viii, 1990.

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Extra resources for Algebraic Geometry: Proceedings of the Conference at Berlin 9–15 March 1988

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Ii). 2) EXAMPLE (with isolated singularities for V). Let f = xyz(x + y + z), n = 2. Then V consists of 4 lines in general position in p 2 and its topology is simple to describe. However, the dimensions of the eigenspaces H'(F)a are more subtle invariants. First we compute explicit bases for the homogeneous components of Q(f): Q(f)o = (l>,Q(f)1 = ,Q(f)2 = Q(f)3 = Q(fh = for k and ~ 4. Then we look for the elements in H2(K') and define: w xy = x(x + 2y + z)dy /\ dz + y(2x + y + z)dx /\ dz and w yz , Wzx by cyclic symmetry.

0 + dim11k - 1 (F)0 = dim Hk(U") + dim 11k-(U). It follows that the first and the last map in the above exact sequence are trivial. 0 as above. 19) Hk + 1 (Cf+ f')o ..... e. W is a sum W 1 + ... 19) is commutative up to a constant. The big rectangle in the diagram is commutative in a similar way by the next result. 20) LEMMA. Ri*b(}1/1 = -l/Nb(}. Proof We have to show that both sides of this equality yield the same result when applied to an element W = w 1 + ... + wp E Hk(Cf)c as above. Since these computations are rather tedious, we treat here only the case p = 2 and let the reader check that the general case is completely similar.

Proof. Let p: pn --+ P be the projection presenting P as the quotient ofpn under the group G(w), the product of cyclic groups of orders Wi. Then J = p*(f) = f(x'O°, .. , x:n) is a homogeneous polynomial of degree N and let U be the complement of the hypersurface J = 0 in pn. 2) for U. To simplify the notation, we assume that w = (1, ... ,1) from the beginning. Then U is smooth and it is easier to describe the construction of the Hodge filtration [24]. Let p: X --+ pn be a proper modification with X smooth, D = p-l(V) a divisor with normal crossings in X and (J = X\D isomorphic to U via p.

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