The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\mathcal{L}$ be an invertible sheaf on $\mathcal{C}$ of relative degree $0$ (i.e., for all $t \in \mathbb{A}^n$, $\deg(\mathcal{L}\_t)=0$, where $\mathcal{L}\_t$ is the restriction of $\mathcal{L}$ to $\mathcal{C}\_t$). Denote by $T$ the subscheme of $\mathbb{A}^n$ parametrizing all points $t \in \mathbb{A}^n$ such that $h^0(\mathcal{L}\_t) \not=0$. Is $T$ regular? If not true in general is there any condition on $\mathcal{C}$ under which this holds true? Any reference/hint will be most welcome.



Variation of global sections of line bundles
For a general smooth base $S$ it is not true that $T$ is regular. Assume even that $\mathcal{C} = C \times S$ for a fixed smooth curve $C$. The line bundle $\mathcal{L}$ then defines a morphism $f \colon S \to Pic^0(C)$ and then $T = f^{-1}(\{0\})$. Conversely, given a morphism $f$ recovers back the line bundle $\mathcal{L}$. Thus, in case of $\mathcal{C} = C \times S$, the scheme $T$ is just the preimage of the origin in $Pic^0(C)$ under an arbitrary map $f \colon S \to Pic^0(C)$.


To construct an example when $T = f^{-1}(\{0\})$ is singular, one can consider the blowup of $Pic^0(C) \times \mathbb{A}^1$ in a point over $\{0\}$. In this case $T$ is even reducible.


