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):

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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?