Let $X\to T$ be a flat familiy of curves of genus $g$ where $T$ is a smooth curve over a field $K$. Assume that for $t_0\in T$ the fiber $X_{t_0}$ is hyperelliptic and for all other $t\in T$ the fiber $X_t$ is not. Is it then always possible to construct a flat family $Y$ of degree $2g-2$ curves in $\mathbb{P}^{g-1}$ over $T$ such that for $t\neq t_0$ the fiber $Y_t$ is $X_t$ canonically embedded? What would $Y_{t_0}$ be in that case? Is it (as a set) the rational normal curve of degree $g-1$?
Degeneration to a hyperelliptic curve
1
$\begingroup$
algebraic-geometry
algebraic-curves
deformation-theory