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Let $f\colon X\to Y$ be a morphism between irreducible $\mathbb{C}$-varieties, such that $f$ is 1-1 and surjective, but $df$ fails to be injective along a subvariety $Z$. (This forces $Y$ to be singular).

My question is, in this case, with the (possible non-reduced) scheme structure, What is the scheme structure of the fiber product $Y\times_{Y\times Y} X\times X=X\times_Y X$ ?

Thanks for the comment @abx. Probably I should be more precise, what I would like to ask is, what is the scheme structure of $X\times_Y X$? For example, the base set is just the diagonal $\Delta_X$ union with the double point set, but as a scheme there should be a non-reduced structure along the non-immersion locus, I suppose. My question was to ask this scheme structure. For example, consider $X,Y\subset \mathbb{C}^4$ parametrized by $X=(x, -6x^2, 8x^3, w)$, and $Y=(3x^4, -6x^2, 8x^3, w)$, with $f\colon (x,y,z,w)\mapsto (x^4+yx^2+zx, y,z,w)$. $X$ is a smooth surface, while $Y$ is not. $f$ is 1-1 and surjective, but not local immersion along $(0,0,0,w)$. In this case what is the scheme structure of $X\times_Y X$?

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    $\begingroup$ The first one is $X$, and the second one $X\times_YX$. This has nothing to do with singularities, or with the special properties of $f$ — just compose the appropriate cartesian diagrams. The question would be a better fit for MSE. $\endgroup$
    – abx
    Commented Sep 18, 2019 at 9:23
  • $\begingroup$ Thanks for the comment. Probably I should be more precise, what I would like to ask is, what is the scheme structure of $X\times_Y X$? For example, the base set is just the diagonal $\Delta_X$ union with the double point set, but as a scheme there should be a non-reduced structure along the non-immersion locus, I suppose. My question was to ask this scheme structure. $\endgroup$
    – Winnie_XP
    Commented Sep 19, 2019 at 1:15
  • $\begingroup$ You will get the diagonal with an embedded component along $Z$; the precise structure depends very much on the particular morphism you choose. You might try first the case where $f$ is the normalization of the cuspidal cubic $y^2=x^3$ to see what happens. Your example is doable in the same way, though more complicated. $\endgroup$
    – abx
    Commented Sep 19, 2019 at 4:07
  • $\begingroup$ @abx Dear abx: Thanks for your reply! that's exactly what I was expecting for, that the fiber product should have an embedded component along Z. (I did the cuspidal curve case, but I struggled in my example since I had trouble finding the kernel for the tensor product of $\mathbb{C}$ algebras. I might will try to hit Macaulay 2, lol.) $\endgroup$
    – Winnie_XP
    Commented Sep 19, 2019 at 4:37

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