Let $X$ be a Fano 3-fold with terminal singularities. Is there some bound (possibly explicit) for the Picard rank of $X$ ?
If $X$ is smooth, it is well-known that the bound is $10$, obtained by del Pezzo fibrations of degree $1$.
With some assumptions on $X$, the bound is $7$ (Nikulin, "On the Picard number of Fano 3-folds with terminal singularities"). Does someone knows what happens in the general case?