On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple ideals

摘要

Let (S, m) be a graded algebra of dimension d generated by finitely many elements of degree 1 over a field k and a homogeneous equimultiple ideal I of S with htI = h > 0. In this paper we will show that if a(1) less than or equal to a(2) less than or equal to (. . .) less than or equal to a(h) is the degree sequence of a minimal homogeneous reduction of I, then the mixed multiplicity e(m([d-i]), I-[i]) = a(1)a(2) (...) a(i)e(S) for all 0 < i < h and the multiplicity of Rees algebra e(R(I)) = [1 + Sigma(i=1)(h-1) a(1)a(2) (. . .) a(i)]e(S). (C) 2003 Elsevier B.V. All rights reserved. [References: 12]