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Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=h\circ g$ where $g:Y\rightarrow X$, $h:X\rightarrow Z$ are two generically finite morphism.

Can we say something about the singularities of $X$ or could it have arbitrarily bad singularities ?

For instance, does the assumptions imply that $X$ has at worst canonical singularities ?

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It can have bad singularities and dimension doesn't matter. Take an arbitrarily singular $X$ of dimension $d$. There is always a finite generic projection $X \to Z = {\mathbb{P}}^d$ (and $Z$ is smooth and Fano). On the other hand let $Y \to X$ be a resolution of singularities.

Then $Y \to X \to Z$ is generically finite (birational followed by finite). In particular, such a factorization exists for any $X$.

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