On the multiplicity and the Cohen–Macaulayness of fiber cones of graded algebras

Let AA be a Noetherian local ring with the maximal ideal mm and an mm-primary ideal JJ. Let S=⨁n≥0SnS=⨁n≥0Sn be a finitely generated standard graded algebra over AA. Set S+=⨁n>0SnS+=⨁n>0Sn. Denote by FJ(S)=⨁n≥0→(Sn/JSn)FJ(S)=⨁n≥0→(Sn/JSn) the fiber cone of SS with respect to JJ. The paper characterizes the multiplicity and the Cohen–Macaulayness of FJ(S)FJ(S) in terms of minimal reductions of S+S+.