Let $f:X \to Y$ be a (flat and projective) family of hypersurfaces over a smooth curve $Y$. If every fiber of $f$ is smooth, it is clear that the total space $X$ is also smooth. I would like to know if the smoothness still holds when we also allow singularities of ordinary double points:
If $f$ has finitely many singular fibers but every singular fiber contains only one ordinary double point, is it still true that $X$ is smooth?
Here an ordinary double point means the tangent cone is nondegenerate. Since the total space of a Lefschetz pencil is always smooth (as blow-up a smooth one along a smooth locus is smooth), I guess it is reasonable to expect smoothness with those nice singular fibers.
Thanks in advance.
Best Answer
No. For instance take $Y = \mathbb{A}^1$ (with coordinate $t$) and set $$ X = \{ xy + t(xz - y^2) = 0 \} \subset \mathbb{A}^1 \times \mathbb{P}^2. $$ The fibers of $X$ over $\mathbb{A}^1 \setminus \{0\}$ are smooth, the fiber over 0 has an ordinary double point, but $X$ is singular (at that point).