# Singularities of fibrations

Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three.

Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}^2\setminus C$ the fiber $f^{-1}(p)$ is a smooth curve and when $p\in C$ then $f^{-1}(p) = A_p\cup B_p$ where $A_p, B_p$ are smooth curves intersecting transversally in a point $x_p = A_p\cap B_p$.

In this situation what could be said on the singularities of $X$? For instance deos this imply that $X$ is singular at most in the points of the form $x_p$ and that the locus of these points is a locus of at worst canonical singularities for $X$ ?

• If $f:X\to \mathbb P^2$ is flat, then (under your assumptions) the morphism $X\to \mathbb P^2$ is smooth at all $x\in X$ such that $x\neq x_p$ with $p\in C$. In particular, $X$ is nonsingular at those points. Apr 14 '17 at 20:51

Your assumptions imply that all the fibers are Gorenstein. Since $\mathbb P^2$ is smooth, this implies that then $X$ is Gorenstein. Furthermore, your assumptions imply that all the fibers are $1$-dimensional, so $f$ is equidimensional, and hence it is flat by "miracle flatness". So, as Ariyan noted, then $f$ is smooth at every $x\in X$ such that $x\neq x_p$ and hence $X$ is smooth at all of those points. This also implies that $X$ is normal (it is $S_2$ since it is Gorenstein and its singular locus has at most codimension $2$, so it is also $R_1$).
Now, looking at one of these $x_p$'s, we still have that $X$ is Gorenstein and we know that a complete intersection curve through this point is a simple node. As an exercise, try to prove directly (say via a direct local computation) that this implies that then $X$ is canonical (of index $1$) at these points.
If you get stuck, then use this argument: Let $C_1$ and $C_2$ be two smooth curves in $\mathbb P^2$ that intersect transversally at $p$ and let $D_i=f^*C_i\subseteq X$ for $i=1,2$. (For the record, the $D_i$ are reduced divisors on $X$.) Now, $D_1\cap D_2 =f^{-1}(p)$, which is a nodal curve, so it has slc singularities. Then by inversion of adjunction (applied twice) $(X,D_1+D_2)$ is log canonical. Now if you scrape away the $D_i$, this actually means that $X$ is terminal at the points of the form $x_p$. (Note that $(D_1, D_1\cap D_2)$ is also log canonical, so $D_1$ is canonical. It is locally isomorphic to a cone over a quadric.)
• Thanks a lot for the answer. In my case $X$ is a complete intersection $3$-fold in $\mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^2$. So $X$ is Gorenstein. I checked that all the fibers of $f$ have dimension $1$ so from what you said $f:X\rightarrow \mathbb{P}^2$ is flat. But I just found a reducible fiber with a non reduced component (of multiplicity 2). How can this happen? In this situation shouldn't all the fibers be reduced? Apr 15 '17 at 17:24
• OK, so your fibers are not exactly as you originally described, but having reduced components in the fibers is not unusual. For instance, if you resolve the indeterminacy of the map $(x,y)\mapsto {y^2}/x$, you get exactly that behavior. Apr 16 '17 at 1:26