Irreducible constant dimensional fibres --> irreducibility?

Let X, Y be affine varieties over an algebraically closed field of
characteristic zero. Assume Y is irreducible. Let f: X -->Y be a
surjective regular morphism with the property that the fibre over each
point of Y is irreducible, and of the same dimension. I was wondering
if:

1) This implies that X is irreducible.


2) If (1) is not true, can we at least claim X has only one
irreducible component of Dim(Y)+Dim (fiber f).

Comment:

If X and Y are projective varieties, then under the assumptions
stipulated in the first 3 lines of this post, X would be irreducible.
The proof unfortunately uses the fact that the image of a projective
variety under a regular map is closed, something that is not the case
for affine varieties in general.



.

Loading