Let $f\colon X\to Y$ be a finite morphism of quasiprojective varieties (I’m most interested in the case that $f$ is normalization and the varieties are over $\Bbb C$). Is the following statement true (at least locally on $Y$)?



> 
> There are embeddings $i$ and $j$ of $X$ and $Y$, resp., into smooth quasiprojective varieties $X’$ and $Y’$, resp., along with a *finite* morphism $f’$ such that $f’\circ i = j\circ f$.
> 
> 
> 



Embedding a finite morphism into a finite morphism of smooth varieties
This is always possible. I will first explain the local construction, and then sketch a proof of how to globalise.


**Local problem.** Let $A \to B$ be a finite map of finite type $k$-algebras. Does there exist a finite map $A' \to B'$ of smooth $k$-algebras admitting a surjection to $A \to B$?


*Construction.* Suppose $B = A[b\_1,\ldots,b\_r]$; then each $b\_i$ satisfies some monic minimal polynomial $f\_i \in A[x\_i]$. This gives a surjection
$$A[x\_1,\ldots,x\_r]/(f\_1,\ldots,f\_r) \twoheadrightarrow B.$$
This map need not be an isomorphism in general, but the result for $A[x\_1,\ldots,x\_r]/(f\_1,\ldots,f\_r)$ implies that for $B$. Thus, we may assume $B = A[x\_1,\ldots,x\_r]/(f\_1,\ldots,f\_r)$.


Now write $f\_i = \sum\_j a\_{ij}x\_i^j$ for some $a\_{ij} \in A$. Let $\phi \colon k[y\_1,\ldots,y\_s] \twoheadrightarrow A$ be any presentation, and adjoin extra variables corresponding to the $a\_{i0}$:
\begin{align\*}
A'=k[y\_1,\ldots,y\_s,z\_1,\ldots,z\_r] &\twoheadrightarrow A\\
y\_i &\mapsto \phi(y\_i),\\
z\_i &\mapsto a\_{i0}.
\end{align\*}
Choose lifts $a'\_{ij} \in A'$ of the $a\_{ij} \in A$ such that $a'\_{i0} = z\_i$ and such that $g\_i = \sum\_j a'\_{ij}x\_i^j$ is again monic. Let $B'$ be the finite extension $A'[x\_1,\ldots,x\_r]/(g\_1,\ldots,g\_r)$. Then $B'$ is the polynomial algebra
$$B' \cong k[y\_1,\ldots,y\_s,x\_1,\ldots,x\_r],$$
since we may eliminate $z\_i = a'\_{i0} = -\sum\_{j > 0} a'\_{ij}x\_i^j$ from the equation $g\_i$. Moreover, we now actually have a pullback:
$$B \cong A \underset{A'}\otimes B',$$
by construction of the $g\_i$. $\square$




---


**Global version.** Recall: if $\mathscr E$ is a locally free sheaf on a scheme $X$, then the *(geometric) vector bundle* $\mathbb V(\mathscr E)$ of $\mathscr E$ is $\operatorname{\underline{Spec}}\_X(\operatorname{Sym}^\*(\mathscr E^\vee))$, viewed as scheme over $X$ through the map $\pi \colon \mathbb V(\mathscr E) \to X$. This gives an equivalence between locally free sheaves and geometric vector bundles; see e.g. [Stacks, [Tag 062N](https://stacks.math.columbia.edu/tag/062N)]¹. If $\mathscr E = \mathscr L$ is a line bundle, we have
$$\pi\_\*\pi^\* \mathscr L \cong \bigoplus\_{i \geq 0} \mathscr L^{1-i},$$
so the section $1 \in H^0(X,\mathscr L^0)$ gives a canonical section $x\_\mathscr L \in H^0(\mathbb V(\mathscr L), \pi^\*\mathscr L)$.


*Construction.* Let $\mathcal O(1)$ be a very ample sheaf on $Y$, and let $\mathscr B = f\_\* \mathcal O\_X$. Then there exists a surjection $\mathcal O(-a)^r \twoheadrightarrow \mathscr B$ for some $a \gg 0$ (i.e. $\mathscr B(a)$ is globally generated for some $a$). Replacing $\mathcal O(1)$ by $\mathcal O(a)$, we may assume $a = 1$ (for convenience).


Set $\mathscr E = \mathcal O(-1)^r$. The surjection $\mathscr E \twoheadrightarrow \mathscr B$ induces a surjection $\operatorname{Sym}^\*(\mathscr E) \twoheadrightarrow \mathscr B$ by the ring structure on $\mathscr B$. For each of the factors $\operatorname{Sym}^\*(\mathcal O(-1)) \to \mathscr B$, the canonical section $x\_i$ satisfies some polynomial equation
$$f\_i = \sum\_{j=0}^{n\_i} a\_{ij} x\_i^j = 0,$$
for some $a\_{ij} \in H^0(Y,\mathcal O(n\_i-j))$. Again, we may replace $\mathscr B$ by $\operatorname{Sym}^\*(\mathscr E)/(f\_1,\ldots,f\_r)$.


On the other hand, we get a locally closed immersion $Y \hookrightarrow \mathbb P^N\_k$ (say given by our very ample line bundle $\mathcal O(1)$). If $\mathscr F = \mathcal O(n\_1) \oplus \ldots \oplus \mathcal O(n\_r)$ on $\mathbb P^N$, then the section $(a\_{10},\ldots,a\_{r0}) \in H^0(Y,\mathscr F|\_Y)$ induces a locally closed immersion
$$Y \hookrightarrow Y'=\mathbb V(\mathscr F) \twoheadrightarrow \mathbb P^N.$$
Consider the coherent $Y'$-algebra given by $\mathscr B' = \operatorname{Sym}^\*(\mathcal O(-1)^r)/(g\_1,\ldots,g\_r)$, where
$$g\_i = \sum\_{j=0}^{n\_i} a'\_{ij}x\_i^j$$
is a monic lift² such that $a'\_{i0} = z\_i$ (the canonical section $z\_{\mathscr O(-1)}$ of $\pi^\* \mathcal O(1)$ on $\mathbb V(\mathcal O(1))$). Then $X' = \operatorname{\underline{Spec}}\_{Y'}(\mathscr B')$ is a smooth $k$-scheme, finite over $Y'$, such that
$$X = X' \underset{Y'}\times Y.$$
This finishes the construction in the global case. $\square$




---


**References.**


[Stacks] A.J de Jong et al, [*The stacks project*](http://stacks.math.columbia.edu).




---


¹The Stacks project (following EGA) does not put a dual in the definition of $\mathbb V(\mathscr E)$. The reason appears to be that duality is not well-behaved in the larger generality of quasi-coherent sheaves. However, the dual version is the 'geometrically correct' version, e.g. it is covariant, and sections of $\pi \colon \mathbb V(\mathscr E) \to X$ are in bijection with $H^0(X,\mathscr E)$.


²There are potential surjectivity issues on global sections, but these can easily be fixed by introducing variables lifting the $a\_{ij}$ like we did for the $a\_{i0}$.


