By Vladimir I. Arnold, Gert-Martin Greuel, Joseph H.M. Steenbrink

In July 1996, a convention was once equipped by means of the editors of this quantity on the Mathematische Forschungsinstitut Oberwolfach to honour Egbert Brieskorn at the get together of his sixtieth birthday. many of the mathematicians invited to the convention were stimulated in a single method or one other via Brieskorn's paintings in singularity thought. It was once the 1st time that such a lot of humans from the Russian tuition might be current at a convention in singularity idea open air Russia. This quantity includes papers on singularity idea and its functions, written through individuals of the convention. in lots of circumstances, they're prolonged types of the talks provided there. the range of matters of the contributions displays singularity theory's relevance to topology, research and geometry, combining rules and strategies from all of those fields, in addition to demonstrating the breadth of Brieskorn's personal pursuits. This quantity includes papers on singularity thought and its purposes, written via contributors of the convention. in lots of situations, they're prolonged types of the talks provided there. the range of topics of the contributions displays singularity theory's relevance to topology, research and geometry, combining rules and strategies from all of those fields, in addition to demonstrates the breadth of Brieskorn's personal pursuits.

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**Example text**

M. L denotes an element of k and (AD: Ad a point of the projective line over k. Now pass to the calculation of R-ideals, which are not 5-ideals. L -=I=- 0 , for i -=I=- 2,7 . L), MjmM = ((I, I), (0, + tD, (0, . L-lt§o. L = 0 , we obtain the following new family: + MjmM = ((I, I), (0, + (0, . Ltl, , 121 R((I, 0), (0, 1)), h2 R((I, (0, 1)). Therefore, we have proved that in this case also par (1, R) = 1.

The list). If val (R) = {4}, then the condition (b) impliesthat R contains an element of the valuation 5 :::; l :::; 7, hence, dominates a SUS of type W. If char k = 2, the condition (e) implies also that R contains an element of the valuation 5: otherwise j = t6RO + k + kt 4 , hence R = t2RO + k and Rj j c:::: Ao. Let now s = 2. If do = 3 then val (R) = {(1,2)} (up to a numbering of the branches; later we omit this notice). Again the condition (d) implies that R contains an element of the valuation (00, l) with 4 :::; l :::; 7, hence, dominates a SUS of type E.

Greuel and G. 4 Let A be a commutative K -algebra with 1 and Leder K(A) an n-dimensional nilpotent K - Lie algebra. Assume that there exist 81 , ... , 8n E Land al, ... ,an E A such that det(8i (aj)) is a unit in A. Let Xl, ... ,Xn be indeterminates and define Fi := exp(-X I 8r) 0·" 0 exp(-Xn8n)(ai). Then COROLLARY is elale over A, the action of L on A lifts to Band H l (L, B) = o. Proof. 3 there exists a faithful representation of L such that HI (L, K[X l , ... , Xn]) = O. Using any such representation, we define the tensor product representation of L on A[X l , ...