I would like to get an understanding of the notion of geometric fibers of the universal family: $$\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}.$$ In fact Knudsen show that family is of the form $$ \mathbf{V}\times\mathbf{Q} \longrightarrow \mathbf{V} \times \mathbb{A}^1$$ where $\mathbf{V} \subset \overline{\mathbf{M}}_{0,n}$ and $\mathbf{Q}=\lbrace (x,y,z)\in \mathbb{A}^3: xy=z \rbrace.$
Moreover, in local coordinate we have $\mathbb{A} \simeq \mathbf{V}.$ Also for n=4, $\overline{\mathbf{U}}_{0,4}$ is the bowup at three point of $\mathbb{P}^1 \times \mathbb{P}^1$ or $\overline{\mathbf{M}}_{0,4} \times_{\overline{\mathbf{M}}_{0,3}}\overline{\mathbf{M}}_{0,4}.$ So, how to show that the geometric fibers of the family are reduced?
