The Stacks project

Exercise 111.52.5. Let $k$ be a field. Let $X$ be a projective, reduced scheme over $k$. Let $f : X \to \mathbf{P}^1_ k$ be a morphism of schemes over $k$. Assume there exists an integer $d \geq 0$ such that for every point $t \in \mathbf{P}^1_ k$ the fibre $X_ t = f^{-1}(t)$ is irreducible of dimension $d$. (Recall that an irreducible space is not empty.)

1. Show that $\dim (X) = d + 1$.

2. Let $X_0 \subset X$ be an irreducible component of $X$ of dimension $d + 1$. Prove that for every $t \in \mathbf{P}^1_ k$ the fibre $X_{0, t}$ has dimension $d$.

3. What can you conclude about $X_ t$ and $X_{0, t}$ from the above?

4. Show that $X$ is irreducible.