Let $\bar{\mathfrak X}\to \mathbb P^N$ be the universal family of hypersurfaces in $\mathbb P^{n+1}$ of degree $d$ and $\mathfrak X \to U$ ($U\subset \mathbb P^N$) be the sub-family of smooth hypersurfaces. The family is a fibration, hence we can talk about the monodromy action $$\rho: \pi_1(U,0)\to {\rm GL}(H^n(\mathfrak X_0,\mathbb Z))$$ where $0\in U$ and $\mathfrak X_0$ is the reference fiber over $0$. We call the image of $\rho$ the monodromy group $G$. (The setting is same as my former question Monodromy on Cohomology?)
Now my questions are:
Can we do the similar things to the moduli space? i.e. if we change the base paramater space to the moduli space $\mathcal M$, is there exist a fibration $\mathfrak Y \to \mathcal M$ such that every fiber is isomorphic to the correspond hypersurface?
If the answer is yes, then we can talk about the monodromy group of the family $\mathfrak Y\to \mathcal M$, say $G'$. A priori $G\subset G'$ because there may be a path in $U$ which is not a loop but becomes a loop after the quotient. Is it true that $G=G'$?
Thanks in advance!