By C. De Concini, C. Procesi (auth.), Francesco Gherardelli (eds.)

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Extra resources for Invariant Theory: Proceedings of the 1st 1982 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held at Montecatini, Italy, June 10–18, 1982

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T~), W(t1, ... ,t~) to be the subspace of sections s E Ho(~,LA) vanishing on 8 1 of order ~ t 1 , •.. 4. 3 we have to discuss the ampleness of (i 1 ... i t ) which has been used there. We start with a general easy fact. Let w,w' denote two distinct Ly+~ fundamental weigths Vw' Vw" Vw+w' the irreducible representations of highest weight w,w' ,w+w'. We have a canonical G equivariantprojection p: Vw ® Vw ' .... 1P (V + ,) of w w ww projective spaces: Remark that1P(V w) x 1P(Vw ') is embedded in P(Vw ® Vw ') via the Segre map.

Nodes as singularities. and Further, any chain of rational curves on which ~)C is trivial has degree 1, Le. consists of a straight line. 1 is toexhibit a I-P - S A of each C which does not satisfy the criteria of the Proposition. general definitions. be a subspace so that ~ SLeW) for We begin with some ~ be a coherent sheaf on a scheme and let Let [G 3 ,M2 1. d ~ lO(g -1). W ~ H~X,~) is generated at each point by the sections in W. 2 A weighted filtration on ~ B is a sequence of subsheaves rl ~ r2 ~ :\ ~:ltj Let ...

Will be called exceptional when rk Pic (X) > R,. 7. REMARK. It is clear from the previous analysis that the main difficulty in computing explicitely the dimensions of the cells lies in T;. 4. In particular for the case of a group variety over G x G we polynomial: Ebiqj, b i = dim r q2R,(w» wEW G considered as a sy~etric have the following computation for the Poincare Hi ( (~,Z): L q2(R,(W)+L(W») (*) wEW (R,(w) the length of w, L(w) the number of simple reflections sa with R,(saw) < R,(w». 8. 1.

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