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.