In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ (over $\mathbb{C}$ ), the monodromy representation is irreducible. Someone has told me that this result is extended by Carlson to local systems obtained as $i$-th eigenspaces of the variation of Hodge structures $(R^{n}g_{*}\mathbb{C}_{Z})i$ of the family of $d$-fold cyclic coverings $g:Z \rightarrow S$ (branched along $X$) for certain values of $d$ and $i$. I could not find the exact result in Carlson's work. Can anyone tell me for which values of $d$ and $i$, the monodromy of eigenspace is irreducible?
In the article "La de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ (over $\mathbb{C}$ ), the monodromy representation is irreducible. Someone has told me that this result is extended by Carlson to local systems obtained as $i$-th eigenspaces of the variation of Hodge structures $(R^{n}g_{*}\mathbb{C}_{Z})i$ of the family of $d$-fold cyclic coverings $g:Z \rightarrow S$ (branched along $X$) for certain values of $d$ and $i$. I could not find the exact result in Carlson's work. Can anyone tell me for which values of $d$ and $i$, the monodromy of eigenspace is irreducible?