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In a one-dimensional local ring R with finite integral closure each nonzerodivisor has a value in N-d, where d is the number of maximal ideals in the integral closure. The set of values constitutes a semigroup, the value semigroup of R. We investigate the connection between the value semigroup and the ring. There is a particularly close connection for some classes of rings, e.g. Gorenstein rings, Arf rings, and rings of small multiplicity. In many respects, the Arf rings and the Gorenstein rings turn out to be opposite extremes. We give applications to evenings, intersection numbers, and multiplicity sequences in the blow-up sequences studied by Lipman. (C) 2000 Elsevier Science B.V. All rights reserved

If R = k[x 1 , . . . , x n ]/I is a graded artinian algebra, then the length of k[x 1 , . . . , x n ]/I s becomes a polynomial in s of degree n for large s. If we write this polynomial as, then the e i 's are called Hilbert coefficients of I. We will study length and Hilbert coefficients of some classes of quadratic algebras.Conjecture 1. If S n /I is a quadratic algebra, then e i (I) ≥ 0 for all i.