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

Completely Controlling the Dimensions of Formal Fiber Rings at Prime Ideals of Small Height