# Non-flat locus for smooth schemes

Let $$X$$, $$Y$$ be connected smooth schemes of finite type over an algebraically closed field of characteristic $$0$$. Let $$f:X\rightarrow Y$$ be a non-birational morphism surjective on the underlying topological spaces. Can the non-flat locus of $$f$$ be non-empty of codimension$$\geq 2$$ in $$X$$? For birational morphism, I belive ZMT plus a purity theorem show that the answer is "no".

P.S. This question was posted on math.stackexchange 3 days ago but there was zero reaction so far.

• @Qfwfq If I understand your example correctly, $X$ is a pushout along inclusions of a closed point in the category of schemes. Then $\mathbb{P}^1$ and $\mathbb{P^1}\times\mathbb{P}^1$ are non-empty closed subschemes of $X$ (stacks.math.columbia.edu/tag/0B7M) whose union is $X$. In particular, $X$ is a reducible scheme. The question explicitly asks for connected smooth schemes over a field (and such schemes can not be reducible, stacks.math.columbia.edu/tag/033M). Unless I got something wrong, your example is not what I am after. Feel free to point out if I got it all wrong. – user137767 Apr 14 at 20:56
• Sorry I misread the question (don't know why I missed the "smooth"). So deleted my previous comment – Qfwfq Apr 14 at 21:54
• Even for birational morphisms, blowing up a point of $Y$ with $\dim Y\geq 2$ has non-flat locus codimension at least 2. The same can be achieved by taking a finite map of smooth varieties and then blowing up a point. – Mohan Apr 15 at 12:08
• @Mohan but we are talking about the non-flat locus in $X$. mathoverflow.net/questions/99591/… – user137767 Apr 15 at 12:50

Suppose that $$f \colon X \to Y$$ is generically finite. Then the locus in $$X$$ where $$f$$ is not flat is the locus where the fiber is positive dimensional.
Now, let $$Z$$ be a projective threefold with a unique singular point $$p \in Z$$, admitting a small resolution $$X \to Z$$ (that is, a proper birational map that is an isomorphism outside of $$p$$, and such that the inverse image $$E$$ of $$p$$ is $$1$$-dimensional). Choose a finite map $$Z \to \mathbb P^3$$; then the composite $$X\to \mathbb P^3$$ is flat precisely outside of $$E$$.
• if I understand correctly, a finite morphism from an integral scheme to $\mathbb{P}^3$ with $f^{-1}(\eta)$ a singleton is an isomorphism. What is the minimum cardinality of the inverse image of the generic point in your examples? Does there exist $Z$ as in your example with a finite morphism to $\mathbb{P}^3$ with $|f^{-1}(\eta)|=2$? – user137767 Apr 16 at 3:22
• The standard example of a threefold $Z$ of this kind is a non-degenerate quadric cone in $\mathbb P^{4}$; this admits a finite morphism of degree 2 to $\mathbb P^{3}$. – Angelo Apr 16 at 23:01