Cohen-Macaulayness of special fiber rings

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $I$ be an $R$-ideal. Inspired by the work of H"ubl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ${\mathcal F}={\mathcal R}/{\mathfrak m}{\mathcal R}$of$I$,where${\mathcal R}$ 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 $G_{\mathrm{\ell }}$ and the expected reduction number, we conclude that $\mathcal{F}$ is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of $\mathcal{R}\mathrm{/}K\mathcal{R}$ for any $\mathfrak{m}$-primary ideal $K$: This result recovers a well-known criterion of Valabrega and Valla whenever $K=I$. Furthermore, we study the relationship among the Cohen-Macaulay property of the special fiber ring $\mathcal{F}$ and the one of the Rees algebra ${\mathcal R}$andtheassociatedgradedring${\mathcal G}$of$I$. Finally, we focus on the integral closedness of $\mathfrak{m}I$. The latter question is motivated by the theory of evolutions.