Let $(X,L)$ be a polarised variety and $\operatorname{Def}(X,L)$ be the Kuranishi space with $o\in \operatorname{Def}(X,L)$ the reference point. Let $(\mathcal{X, L})$ be the Kuranishi family.
Assume that $L$ is very ample over X. I would like to know if the following holds :
There exists a neighbourhood $U$ of $o$ in $\operatorname{Def}(X,L)$ such that for any $t\in U$, $\mathcal L_t$ is very ample on $\mathcal X_t$.
If the above is not true, are there some easy conditions we can impose to make sure the above holds? For example, if we assume furthermore that for $t$ in a neighbourhood of $o$, $\mathcal L_t$ is point-free (generated by globally sections), will the above assertion then hold ?