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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.

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  • $\begingroup$ @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. $\endgroup$
    – user137767
    Commented Apr 14, 2019 at 20:56
  • $\begingroup$ Sorry I misread the question (don't know why I missed the "smooth"). So deleted my previous comment $\endgroup$
    – Qfwfq
    Commented Apr 14, 2019 at 21:54
  • $\begingroup$ 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. $\endgroup$
    – Mohan
    Commented Apr 15, 2019 at 12:08
  • $\begingroup$ @Mohan but we are talking about the non-flat locus in $X$. mathoverflow.net/questions/99591/… $\endgroup$
    – user137767
    Commented Apr 15, 2019 at 12:50

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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 $E$.

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  • $\begingroup$ 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 $|f^{-1}(\eta)|=2$? $\endgroup$
    – user137767
    Commented Apr 16, 2019 at 3:22
  • $\begingroup$ 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 $\mathbb P^{3}$. $\endgroup$
    – Angelo
    Commented Apr 16, 2019 at 23:01

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