## Reference/Known results on the singular behaviour of the fibres of a holomorphic map between compact Kähler manifolds

I have been interested in the following situation of late: Let $$X$$ and $$Y$$ be compact Kähler manifolds with $$\dim_{\mathbb{C}}(Y) < \dim_{\mathbb{C}}(X)$$ and let $$f : X \to Y$$ be a surjective holomorphic map with connected fibres. Let $$S = \{ s_1, …, s_k \}$$ denote the critical values of $$f$$, which is a subvariety of $$Y$$.

I cannot find a detailed account of how bad the singular behaviour of the fibres of $$f$$ can be. For example, do the fibres contain $$(-1)$$ curves (i.e., curves with self-intersection number $$-1$$) or $$(-2)$$ curves?

If anyone can provide references where I can get a better understanding of this, that would be tremendously appreciated.

## Flat Family of Fibres

I have a question about following argument used in an example in Hartshorne’s “Algebraic Geometry” (see page 259):

We have a surjective morphism $$f: X \to Y$$ between schems where $$X$$ is integral and $$Y$$ a nonsingular curve.

My question is why and how to see that there conditions suffice to see that this family is flat?

I guess that in this context a flat family means just that $$f$$ is flat in each fiber, correct?

Especially I don’t see how the condition that $$f$$ is surjective – a purely set theoretical condition – is used to show this algebraic condition?