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 actually 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 constructible, 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.

Update: I have found a lemma for it, which is [SGA3, Expose VI B, Corollary 5.5], it says that:

Let $G$ be a group scheme over $S$, if $G$ is locally of finite presentation and universally open over $S$ with connected fibres, then $G$ is separated and of finite presentation over $S$.

The universally openness can be replaced by flatness because "locally of finite presentation" + "flat" imply "universally open", see tag 01UA.

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