Completely Controlling the Dimensions of Formal Fiber Rings at Prime   Ideals of Small Height
  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 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. 
Sarah M. Fleming, Lena Ji, S. Loepp, Peter M. McDonald, Nina Pande,   and David Schwein
math.AC