TITLE: How small can the Mumford-Tate group of hypersurface be?
QUESTION [6 upvotes]: Is there some way  of giving a lower bound on the dimension of the Mumford-Tate group of a hypersurface? Let's say it's of general type, say, of degree $10$ inside $ \mathbb{P}^3$. 
(Edited from here onward, because I forgot about Fermat hypersurfaces):
I would expect small Mumford-Tate groups to be rare. 
For example, a Fermat hypersurface has Mumford-Tate group a torus. Is it possible to say, for example, that points with toral Mumford-Tate group are not Zariski-dense?

REPLY [4 votes]: For hypersurfaces in $\mathbb P^2$, i.e. curves, this follows from the Andre-Oort conjecture for $\mathcal A_g$, $g=(d-1)(d-2)/2$, as long as $d>4$. The moduli space of hyper surfaces is a sub variety of the moduli space of abelian varieties. By Andre-Oort, if the CM points are dense then it must be a special sub variety, that is, a Shimura variety. But the monodromy group of $H^1$ is full symplectic, so if it is a Shimura variety it is all of $\mathcal A_g$, which only happens for $d=3,4$ by dimension counting.
For higher dimension hyper surfaces I would imagine this follows from an Andre-Oort-like statement, although the moduli space of Hodge structures would not usually be a Shimura variety.