Consider the variation of mixed hodge structures which generates at the origin:
$$
f:X = \text{Proj}\left( \frac{\mathbb{C}[t][x,y,z]}{(xy(x + y + tz))} \right) \to \mathbb{A}^1\_t
$$
How can I compute the monodromy of the cohomology groups $\mathbf{R}f\_\*(\underline{\mathbb{Q}}\_X^{\text{Hdg}})$?



How can I determine the monodromy of this variation of mixed hodge structures?
This monodromy is trivial because on the set $t\neq 0$ you can make a change of variables $(x,y,z)\to(x,y,z t^{-1})$, and your ideal becomes $x y (x+y+z)$, so it doesn't depend on $t$. So your family is just a constant family.


