Local cohomology of bigraded Rees algebras and normal Hilbert coefficients

Let (R, m) be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and (I) over bar be the integral closure of an ideal I in R. Necessary and sufficient conditions are given for (Ir+1 J(s+1) ) over bar = a (Ir+1 J(s+1)) over bar + b (Ir+1 J(s)) over bar to hold for all r >= r(0) and s >= s(0) in terms of vanishing of [H-(at1bt2)(2)((R') over bar (I.J))](r(0),s(0)), where a is an element of I, b is an element of J is a good joint reduction of the filtration {(I-r J(s)) over bar}. This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing of (e) over bar (2)(I J) is shown to be equivalent to Cohen-Macaulayness of (R) over bar.(I, J). (C) 2013 Elsevier B.V. All rights reserved