## 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):

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?