the deformation of a fibration V.S. the deformation of the generic fiber

Question

Let $X \to C$ be a flat family over a curve. A deformation of the fibration gives a deformation of the generic fiber $X_\eta$. Now my questions are the following.

1. If we have a deformation of the generic fiber, is there any way to ``lift" the deformation to the fibration?

2. A slightly weaker version: if we have a deformation of the generic fiber, is it possible to choose one model of $X \to C$ such that the deformation is induced by this fiberation.

Based on the answer given, let me add that a general fiber is rationally connected or even fano. And if you wish, the base is a rational curve. Also the total space should be smooth.

Answer 1

The answer is no.

The following example is typical. Take a surface of general type $X$ which is obtained as

$X:=(C_1 \times C_2) /G$ where

• $C_1$, $C_2$ are smooth curves
• $G$ is a finite group acting faithfully on both $C_1$ and $C_2$ and freely on their product (the action is the diagonal one).
• $C_i/G \cong \mathbb{P}^1$ and $f_i \colon C_i \to C_i/G$ is branched in three points.

(F. Catanese called these surfaces Beauville surfaces, there are many of them).

Then there are two natural isotrivial fibrations

$g_i \colon X \to C_i/G \cong \mathbb{P}^1$.

On the other hand, one proves that $X$ is a rigid scheme, so it has no abstract deformations at all. In particular no non-trivial deformation of the general fibre of $g_i$ can lift to a deformation of the fibration.