# Examples of flat projective morphisms with non-divisorial branch locus

What are some interesting examples of flat projective maps $$f : X \to Y$$ between smooth varieties such that the discriminant locus $$\Delta = \{ y \in Y \mid X_y \text{ is singular} \} \subset Y$$ has a component of codimension $$\ge 2$$?

I am particularly interested in an example with $$Y$$ a smooth proper surface and $$\Delta$$ a finite collection of points.

When $$X$$ is not assumed to be smooth, some examples are given in this stacks project blog post.

I believe the following example is due to Mumford, but I don't have a reference at hand. Let $$S$$ be a smooth projective surface and $$p$$ a closed point of $$S$$. Let $$b: X\rightarrow S\times S$$ be the blowing up of the union of $$S\times \{p\}$$ and the diagonal of $$S\times S$$, and let $$f:X\rightarrow S$$ be the composition of $$b$$ and the first projection. Then $$X$$ is smooth, $$f$$ is smooth outside $$p$$, but $$f^{-1}(p)$$ is the union of $$b^{-1}(p,p)$$ and the strict transform of $$\{p\}\times S$$. $$f$$ is equidimensional, hence flat.
• Why is $X$ smooth? That union is not Cohen-Macaulay . . . Commented Aug 14 at 9:45
• @Jason Starr: A local model is the map $b:\mathbb{A}^4\rightarrow \mathbb{A}^4$, $b(x,u,z,v)=(x,y,z,t)$ with $y=xu$, $t=zv$. This blows up the planes $x=y=0$ and $z=t=0$.
• @JasonStarr: This blowup can be realized either as the blowup of one component followed by the blowup of the strict transform of the other (in any order), or by first blowing up the intersection point, then the two strict transforms of the components, and then contracting back the strict transform of the first exceptional divisor (isomorphic to the blowup of $\mathbb{P}^3$ in a pair of skew lines) to $\mathbb{P}^1 \times \mathbb{P}^1$. Commented Aug 15 at 8:42