Completely controlling the dimensions of formal fiber rings at prime ideals of small height

Abstract

Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension $3$ or greater. Assuming that $|T| = |T/\mathfrak m|$, where $\mathfrak m$ is the maximal ideal of $T$, we construct a local UFD $A$ whose completion is $T$ and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, $T$ is regular and has characteristic zero, we can construct $A$ to be excellent.