The Stacks project

Exercise 111.37.4. Give an example of a morphism of schemes $f : X \to {\mathbf A}^1_{\mathbf C} = \mathop{\mathrm{Spec}}({\mathbf C}[T])$ such that the (scheme theoretic) fibre $X_ t$ of $f$ over $t \in {\mathbf A}^1_{\mathbf C}$ is (a) isomorphic to ${\mathbf P}^1_{\mathbf C}$ when $t$ is a closed point not equal to $0$, and (b) not isomorphic to ${\mathbf P}^1_{\mathbf C}$ when $t = 0$. We will call $X_0$ the special fibre of the morphism. This can be done in many, many ways. Try to give examples that satisfy (each of) the following additional restraints (unless it isn't possible):

1. Can you do it with special fibre projective?

2. Can you do it with special fibre irreducible and projective?

3. Can you do it with special fibre integral and projective?

4. Can you do it with special fibre smooth and projective?

5. Can you do it with $f$ a flat morphism? This just means that for every affine open $\mathop{\mathrm{Spec}}(A) \subset X$ the induced ring map $\mathbf{C}[t] \to A$ is flat, which in this case means that any nonzero polynomial in $t$ is a nonzerodivisor on $A$.

6. Can you do it with $f$ a flat and projective morphism?

7. Can you do it with $f$ flat, projective and special fibre reduced?

8. Can you do it with $f$ flat, projective and special fibre irreducible?

9. Can you do it with $f$ flat, projective and special fibre integral?

What do you think happens when you replace ${\mathbf P}^1_{\mathbf C}$ with another variety over ${\mathbf C}$? (This can get very hard depending on which of the variants above you ask for.)