Weight filtration and generating level

We study the canonical mixed Hodge module structure associated to the $\mathscr{D}_X$-module $\mathscr{M}(f^{-\alpha}):=\mathscr{O}_X(*f)f^{-\alpha}$. We particularly focus on the weight filtration and extend many known results to the weighted setting. We obtain new relations between Hodge theory and birational geometry. We derive a general formula for the Hodge and weight filtrations on $\mathscr{M}(f^{-\alpha})$, and use this to obtain results concerning the largest weight of $\mathscr{M}(f^{-\alpha})$ and the generating level of weight filtration steps. Finally, we obtain expressions for several classes of divisor, including certain parametrically prime divisors.