By Leonard Lovering Barrett

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As a matter of fact, Mumford was partly motivated by later work of Hirzebruch on cusp singularities which may be seen as a natural continuation of his thesis. Let T be the standard complex algebraic torus C*2 C (02. The basic fact is that Xn,q contains the algebraic torus Tn,q = T/C'n,q. We shall ,,,,q- construct the resolution Xn,q 4 Xn,q by gluing several copies of C2 which map to Xn,q so that T is mapped isomorphically onto Xn,q C C3 be the Weierstrahspace given by the equation :T = 0. Let Xn,q -4 Xn,q be the normalization reap induced by the map C2 -- Xn,q given by (zl, z2i z3) = (ti, t2, tt,--`I).

In their paper "Komplexe Rdume", published 1958 in Mathematische Annalen they proved that the notions of complex space in the sense of Behnke and Stein and in the sense of Cartan were coextensive. Grauert and Remmert also clarified a question that Hirzebruch had to leave unanswered in his thesis. They proved that every k-dimensional normal complex space can be presented locally as an algebroid covering of a domain in Ck. This means that locally it is the normalization of a Weierstraf3covering defined by an irreducible Weierstral3polynomial 24 EGBERT BRIESKORN in C{z1,...

V. Berry & J. M. Robbins, Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proc. R. Soc. London A 453 (1997) 1771-1790. [3] A. Borel & F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958) 458-538. [4] G. I. Lehrer, On the Poincdre series associated with Coxeter group actions on complements of hyperphones, J. London Math. Soc. (2) (1987) 275-294. DEPARTMENT OF MATHEMATICS & STATISTICS UNIVERSITY OF EDINBURGH SURVEYS IN DIFFERENTIAL GEOMETRY, 2000 Vol.