# Examples of semi-abelian schemes over a curve

Let $$C$$ be a nice curve, i.e. $$C$$ is a smooth, projective, geometrically integral scheme of dimension $$1$$ over a field $$k$$. For example, (assuming the characteristic of $$k$$ is neither 2 or 3) an elliptic curve defined by $$y^2z-x^3-axz^2-bz^3=0$$ where $$\Delta=-16(4a^3+27b^2)\neq 0$$.

And I want to know about examples of constructing semi-abelian scheme $$\mathcal{A}$$ over $$C$$, of course I want them to not be abelian schemes over $$C$$. I know that a semi-abelian scheme over $$C$$ is like a family of semi-abelian varities over the fibres. When all the fibres are abelian varieties, then it is an abelian scheme. I want to calculate the set of bad points (in the sense the fibres are not abelian) in the example as well, to understand how far from being abelian it is.

In particular, I want the generic fibre to be good (i.e. $$\mathcal{A}_{\eta}$$ is an abelian variety where $$\eta$$ is the generic point of $$C$$).

• Just take $\underline{\operatorname{Pic} }^{\mathrm{o}}(X/C)$ for a semi-stable family of curves $X\rightarrow C$.
– abx
Aug 26, 2021 at 6:56
• Or, as a concrete example, take a Weierstrass equation with coefficients in $k(C)$ such that removing the singular point in the bad fibres is already the Neron model, e.g. only bad fibres of type I$_{1}$. Feb 2 at 9:58