Cohen-Macaulayness of special fiber rings

Let (R, m) be a Noetherian local ring and let I be an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ℱ = script R sign/mscript R sign R of I, where script R sign denotes the Rees algebra of I. Our key idea is to require 'good' intersection properties as well as 'few' homogeneous generating relations in low degrees. In particular, if I is a strongly Cohen-Macaulay R-ideal with G_ℓ and the expected reduction number, we conclude that ℱ is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of script R sign/Kscript R sign for any m-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K=I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ℱ and the Cohen-Macaulay property of the Rees algebra script R sign and the associated graded ring script g sign of I. Finally, we focus on the integral closedness of mI. The latter question is motivated by the theory of evolutions.