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

ISBN-10: 9400906854

ISBN-13: 9789400906853

ISBN-10: 9401067937

ISBN-13: 9789401067935

The convention on Algebraic Geometry, held in Berlin 9-15 March 1988, used to be organised by means of 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 chain organised by way of the Humboldt-Universitat at common periods of 2 or 3 years, with the aim of supplying a gathering position for mathematicians from japanese and western international locations. the current quantity comprises gildings of a part of the lectures awarded on the convention and a few articles on comparable topics. All papers have been topic to the general refereeing process of Compositio Mathematica, and H. Kurke acted as a visitor editor of this magazine. The papers concentrate on genuine topics in algebraic geometry and singularity concept, corresponding to vector bundles, arithmetical algebraic geometry, intersection idea, moduli and Hodge idea. we're thankful to all those that, via 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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**Example text**

First we apply A to the equality (i) and get (iv) w 2/(f - 1) + w 2 = df /\ A(W2)/(N(f - 1)). Next we can divide this equality by (f - 1)' and get (v) w 2/(f - 1y+l + w 2/(f - 1)' = -d(A(w2)/Ns(f - 1)S) + dA(w 2)/Ns(f -)'. If we apply A to (ii), we get This should be put in (v), one should apply once more this trick getting a term containing dA(w 1 ) and then replace this by qWl as follows by applying A to (iii). Let As = wds(f - 1)' + w 2/(f - 1y+ 1 and note that As is a closed form on C"+ 1 \F for any s ~ 1.

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 =

7). Then X(V) = Xm(V(w, N)) + (_1)n L dim M(gat N- w X(F) = Xm(F(w, N)) + (_1)n+1 L L aeZ aeZ j= 1,N dimM(ga)(m-1)N-w+j for all m large enough. When all the singularities ga are weighted homogeneous, it is enough to take m ~ n + 1. Proof On a formal level, note that the formulas in (i) are a special case of the formulas in (ii), obtained by taking m divisible by all ka = 1Gal, a E Z. The proof of (i) is purely topological and independent of our previous results. Let a, H, ... , be as above. We may takef' close enough to f such that for all aE Z the intersection Fa = B.

### Algebraic Geometry: Proceedings of the Conference at Berlin 9–15 March 1988 by F. A. Bogomolov, A. N. Landia (auth.), H. Kurke, J. H. M. Steenbrink (eds.)

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