Effectivity for bounds on points of good reduction in the moduli space of hypersurfaces

Let $S$ be a finite set of primes. For sufficiently large $n$ and $d$, Lawrence and Venkatesh proved that in the moduli space of hypersurfaces of degree $d$ in ${\mathbb{P}}^n$, the locus of points with good reduction outside $S$ is not Zariski dense. We make this result effective by computing explicit values of $n$ and $d$ for which this statement holds. We accomplish this by giving a more precise computation and analysis of the Hodge numbers of these hypersurfaces and check that they satisfy certain bounds.