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

Abstract

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 ht I= h>0. In this paper we will show that if a 1⩽ a 2⩽⋯⩽ a h is the degree sequence of a minimal homogeneous reduction of I, then the mixed multiplicity e( m [d−i],I [i])=a 1a 2⋯a ie(S) for all 0< i< h and the multiplicity of Rees algebra e(R(I))=[1+∑ i=1 h−1 a 1a 2⋯a i]e(S) .