Minimal Monomial Reductions and the Reduced Fiber Ring of an Extremal Ideal

Let $I$ be a monomial ideal in a polynomial ring $A=K[x_1,...,x_n]$. We call a monomial ideal $J$ to be a minimal monomial reduction ideal of $I$ if there exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction ideal of $I$. We prove that there exists a unique minimal monomial reduction ideal $J$ of $I$ and we show that the maximum degree of a monomial generator of $J$ determines the slope $p$ of the linear function $\reg(I^t)=pt+c$ for $t\gg 0$. We determine the structure of the reduced fiber ring \mathcal{F}(J)_{\red} of $J$ and show that \mathcal{F}(J)_{\red} is isomorphic to the inverse limit of an inverse system of semigroup rings determined by convex geometric properties of $J$.