Let $f:X\to Y$ be a (flat and projective) family of hypersurfaces over a variety $Y$, and let $B\subset Y$ be a closed subset with ${\rm codim_Y(B)}\geq 2$. I would like to know that:
if we know (the isomorphism class of) every fiber in $Y\backslash B$, is it true that we also know the (isomorphism class of) fiber in $B$?
If we drop out the condition of codimention $2$, then we have many counter-examples, say $x^2-y^2=t$ defines a family over $t\in \mathbb A^1$, and all fibers except at $0$ are isomorphic. But this phenomenon does not occur when codimension of $B$ is at least $2$.