Let $X$ be a non singular complex projective surface (Hartshorne notation!) and consider a morphism $f:X\longrightarrow\mathbb P^1_{\mathbb C}$ with the following properties:

  1. $f$ is flat
  2. $f$ is proper
  3. The fibers of $f$ are connected

Under these conditions can I conclude that the fibers of $f$ are all reduced? And what we can say about the irreducibility of the fibers?


First of all, note that under your conditions on $X$, the proper hypothesis is redundant; moreover, the flat hypothesis is equivalent to $f$ being surjective (e.g. by Hartshorne Proposition III.9.7, or preferably Exercise III.10.9).

Now let $\phi : \mathbf P^2 \dashrightarrow \mathbf P^1$ be the rational map $[x,y,z] \mapsto [x^2,yz]$. By blowing up points of $\mathbf P^2$ (possibly repeatedly) we can turn this into a surjective morphism $f: X \rightarrow \mathbf P^1$ from a smooth projective surface. The fibres of $f$ are which are smooth at the basepoints $[0,1,0]$ and $[0,0,1]$ of $\phi$ are isomorphic to the subschemes $C_{\lambda, \mu}$ of $\mathbf P^2$ defined by

$$C_{\lambda,\mu} = \{ [x,y,z] \mid \lambda x^2 + \mu yz =0 \}.$$

(for the appropriate value of $[\lambda, \mu] \in \mathbf P^1$).

Note that $C_{0,1}$ is a union of two lines, therefore reducible.

As for the fibre of $f$ over the point $[0,1] \in \mathbf P^1$: it will contain $C_{1,0}$ as a non-reduced irreducible component, although I think it will have extra components.

For an example with an irreducible but non-reduced fibre one has to do a little more work, since these kinds of blowup examples don't work. One class of examples is Enriques surfaces: these always have a morphism to $\mathbf P^1$ with precisely two double fibres.


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