This question is related to my previous question: Singularities of fibrations

Assume that $X$ is a complete intersection irreducible $3$-fold in a product of projective spaces. So that $X$ is Gorenstein. Let $f:X\rightarrow \mathbb{P}^2$ be a morphism with connected fibers such that $f^{-1}(p)$ is a smooth curve for any $p\in\mathbb{P}^2\setminus D$, where $D\subset\mathbb{P}^2$ is a smooth curve. Now, suppose that for $p\in D$ any irreducible component $C_i$ of $f^{-1}(p)$ has multiplicity at most $2$ (we could have non reduced components), and we could have also more than two components of a fiber passing through the same point.

If I got right the answer of the previous question by Sándor Kovács then $f$ is flat and $X$ is smooth outside the non reduced components of the fibers and of the intersection points of the components of a fiber, let $U\subseteq X$ be the open subset defined by these conditions.

Then $X$ could be singular in the points $x\in X\setminus U$. In this situation, which seems pretty desperate to me, could we still say something about the singularities of $X$ of could they be arbitrarily bad ?


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