Let $X$ be a scheme over $\mathbb C$, such that $A=X_{\textrm{red}}$ is an abelian variety. Suppose the tangent spaces to $X$ are all of dimension $\dim X+1$. Then $X$ is non-reduced everywhere, but I cannot figure out what kind of non-reducedness can actually occur. So here is my question:
Q. Is $X$ isomorphic to $A\times \textrm{Spec }\mathbb C[\epsilon]$?
In general, one can produce non-trivial "ribbons" of, for instance, lines in $\mathbb P^3$. For smooth group schemes I would think there is only the trivial non-reduced structure, but I cannot prove it. Thank you for sharing any thought!
I believe an elliptic curve over $\mathbb{C}[\epsilon]$ whose $j$-invariant is not in $\mathbb{C}$ gives what you want.