Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?

Let $ \pi:\mathcal{A}\rightarrow C$ be a semi-abelian scheme, i.e. $ \mathcal{A}$ is a smooth separated commutative group scheme over $ C$ via $ \pi$ with geometrically connected fibres, such that each fibre $ \mathcal{A}_v$ , where $ v\in C$ , is an extension of an abelian variety $ \mathcal{B}_v$ by a torus $ T_v$ over the residue field $ \kappa(v)$ , or equivalently it is a semi-abelian variety.

I would like to ask if the set $ \{v\in C:\mathcal{A}_v\textrm{ is an abelian variety}\}$ is open. In particular where $ C$ is a smooth, projective, geometrically integral curve over a perfect field of characteristic $ p$ , and the set contains the generic point of $ C$ .

I would like to use the constructibility results from EGA, but all of them relies on the condition that “the morphism is of finite presentation”, which I can’t show. To be clear, it is acutally the property of “quasi-compactness” of $ \pi$ which I can’t show, with quasi-compactness then I can show finite presentation hence the above set is consturctible, and it contains the generic point so it contains a dense open subset. On a curve its complement is a finite set of closed points, so itself must be open.

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?