The conductor of one-dimensional gorenstein rings in their blowing-up

Abstract

Let (A,M) be a local, one-dimensional, Cohen-Macaulay ring of multiplicity e=e(A)>1 and Hilbert function H(A). Let I=Ann_A (B/A) be the conductor of A in its blowing up B. Northcott and Matlis have proved that if the embedding dimension emdim A of A is 2 then I=M^e−1 [3; Corollary 13.8]. If emdim A>2 little is know about I. In [6] and [7] I is computed when the associated graded ring G(A) is reduced (in this case B in the integral closure of A). In this paper we compute I when A is Gorenstein. There are in general upper and lower bounds for I in terms of a power of M and we start discussing when these bounds are attained. In particular we show that in the extremal situation I=M^e−1 one has emdimA=2 (thus inverting the result of Northcott and Matlis). Then we consider the case of Gorenstein rings. We prove that if G(A) in Gorenstein then I=M^ϑ where ϑ=Min{n‖H(n)=e}. If more generally A is Gorenstein then I⊂M² or emdim A=e(A)=2. When A is the local ring of a curve at a singular point p we get, as a consequence of this last result a proof of the following conjecture of Catanese which has interesting geometric applications [1]: if the conductor J of A in its normalization is not contained in M² then p is a node.