Counterexamples to countable-section $\varPi^1_2$ uniformization and   $\varPi^1_3$ separation
  We make use of a finite support product of the Jensen minimal $\varPi^1_2$ singleton forcing to define a model in which $\varPi^1_2$ Uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which Separation fails for $\varPi^1_3$. 
Vladimir Kanovei and Vassily Lyubetsky
math.LO