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.