Critical loci have small images in characteristic zero

Setting. By [F2] the structure morphisms of $X$ and $Y$ are smooth, so every point of either variety has an affine neighbourhood carrying a standard smooth presentation [F3]; by [F4] the classical points of $X$ are its closed points and have residue field $k$, and by [F5] the classical varieties have the affine-model topology with residue field $k$, principal opens forming a basis. For a classical point $x$ of $X$ the differential $d_{x}f$ is the dual of the induced cotangent map [F10] and $T_{x}X$ is the finite-dimensional dual of ${\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$ [F11]. Define $C_r=\{x:\mathrm{rank}{d}_{x}f\le r\}$. We prove (1) that $C_r$ is closed in $X$, and (2) that $\dim \overline{f(C_r)}\le r$.

Charts at a prescribed point. Let $x_0\in X$. The structure morphism of $X$ is smooth, so [F2] provides an affine neighbourhood of $x_0$ whose coordinate ring is standard smooth after a principal shrinking; [F6] supplies finite affine covers of $X$ and of $Y$, [F5] lets us shrink to a principal open contained in any prescribed open neighbourhood, [F7] makes principal opens affine, and [F3] absorbs the principal shrinking into the presentation. Choose in this way an affine chart $V=\mathrm{Spec}C\subseteq Y$ containing $f(x_0)$, where $C=k[V]$ is standard smooth over $k$, and an affine chart $U=\mathrm{Spec}B\subseteq X$ containing $x_0$ with $f(U)\subseteq V$ and $B=k[U]$ standard smooth over $k$; write $B\cong (k[x_1,\dots ,x_n]\mathrm{/}(F_1,\dots ,F_c){)}_g$ with leading $c\times c$ Jacobian minor mapping to a unit of $B$, and $d=n-c$. Let $y_1,\dots ,y_m\in C$ be coordinate classes generating $C$ as a $k$-algebra [F9].

Free differentials on the chart. By [F13] the module ${\mathrm{\Omega }}_{B\mathrm{/}k}$ is the cokernel of the transposed Jacobian map $B^c\to B^n$ of the presentation of $B$, and since the leading $c\times c$ minor is invertible, [F14] shows that this cokernel is free with $B$-basis the images $\mathrm{d}{u}_{c+1},\dots ,\mathrm{d}{u}_n$ of the differentials of the free coordinates $u_{c+1},\dots ,u_n\in B$. For a closed point $x\in U$, [F12] identifies $C_{x}U={\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$ with ${\mathrm{\Omega }}_{B\mathrm{/}k}{\otimes }_B\kappa (x)$ naturally, so that $T_{x}U\cong ({\mathrm{\Omega }}_{B\mathrm{/}k}{\otimes }_B\kappa (x){)}^{\vee }$ by [F11], and the open immersion $U\hookrightarrow X$ identifies $T_{x}U$ with $T_{x}X$ and $d_x(f{\mathrm{\mid }}_U)$ with $d_{x}f$ [F10].

The pullback matrix. Let $\psi \text{\hspace{0.17em}⁣}:C\to B$ be the $k$-algebra map induced by $f{\mathrm{\mid }}_U\text{\hspace{0.17em}⁣}:U\to V$, under the correspondence of [F8]. By [F15] there is a canonical $B$-linear functoriality map $\gamma \text{\hspace{0.17em}⁣}:B{\otimes }_C{\mathrm{\Omega }}_{C\mathrm{/}k}\to {\mathrm{\Omega }}_{B\mathrm{/}k}$, $b\otimes \mathrm{d}c\mapsto b\mathrm{d}(\psi c)$; by [F13] the differentials $\mathrm{d}{y}_1,\dots ,\mathrm{d}{y}_m$ generate the $C$-module ${\mathrm{\Omega }}_{C\mathrm{/}k}$, so their images generate $B{\otimes }_C{\mathrm{\Omega }}_{C\mathrm{/}k}$. Since $\mathrm{d}{u}_{c+1},\dots ,\mathrm{d}{u}_n$ is a $B$-basis of ${\mathrm{\Omega }}_{B\mathrm{/}k}$ [F14], there are unique regular functions $a_{ij}\in B$ with $\mathrm{d}(\psi y_j)={\sum }_{i=c+1}^n{a}_{ij}\mathrm{d}{u}_i(c<i\le n,1\le j\le m);$ denote by $A=(a_{ij})$ the resulting $(n-c)\times m$ matrix over $B$.

Rank identity on the chart. For every closed point $x\in U$ with $y=f(x)$ one has $\mathrm{rank}{d}_{x}f=\mathrm{rank}A(x)$, where $A(x)$ is the matrix over $\kappa (x)=k$ obtained by reducing the entries of $A$ modulo ${\mathfrak{m}}_x$; more precisely the two ranks equal the rank of the fibre ${\gamma }_x=\gamma {\otimes }_B\kappa (x)$. Indeed, the cotangent map ${\bar{\psi }}_x^{\mathrm{♯}}\text{\hspace{0.17em}⁣}:C_{y}V\to C_{x}U$ corresponds under the natural isomorphisms of [F12] to ${\gamma }_x\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{C\mathrm{/}k}{\otimes }_C\kappa (y)\to {\mathrm{\Omega }}_{B\mathrm{/}k}{\otimes }_B\kappa (x)$ — the source is identified with $(B{\otimes }_C{\mathrm{\Omega }}_{C\mathrm{/}k}){\otimes }_B\kappa (x)$ by [F16] — and $d_x(f{\mathrm{\mid }}_U)$ is the dual of ${\bar{\psi }}_x^{\mathrm{♯}}$ by [F10], hence has the same rank as ${\bar{\psi }}_x^{\mathrm{♯}}$ by [F17] because these spaces are finite-dimensional [F11]; moreover ${\gamma }_x(\mathrm{d}{y}_j\otimes 1)=\mathrm{d}(\psi y_j)\otimes 1={\sum }_i{a}_{ij}(x)(\mathrm{d}{u}_i\otimes 1)$, the elements $\mathrm{d}{y}_j\otimes 1$ generate the source over $k$ [F13], and $(\mathrm{d}{u}_i\otimes 1{)}_{i>c}$ is a $k$-basis of the target [F14], so by [F18] the rank of ${\gamma }_x$ is the rank of the matrix $A(x)$; finally $\mathrm{rank}{d}_x(f{\mathrm{\mid }}_U)=\mathrm{rank}{d}_{x}f$ because $U\hookrightarrow X$ and $V\hookrightarrow Y$ are open immersions inducing tangent isomorphisms [F10].

Decomposition into components. As a closed subvariety of $X$, the set $C_r$ is Noetherian with finitely many irreducible components $Z_1,\dots ,Z_s$ [F6]; each $Z_i$ is an irreducible closed subvariety of $X$, hence an irreducible classical variety. The image $f(Z_i)$ is irreducible as the continuous image of an irreducible space [F32], so its closure $W_i:=\overline{f(Z_i)}$ in $Y$ is an irreducible closed subvariety of $Y$, i.e. an irreducible classical variety [F31]; the induced morphism $f{\mathrm{\mid }}_{Z_i}\text{\hspace{0.17em}⁣}:Z_i\to W_i$ is dominant because $f(Z_i)$ is dense in $W_i$ [F29].

Generic surjectivity on a component. Fix $i$. By [F28] applied to the dominant morphism $f{\mathrm{\mid }}_{Z_i}$ of irreducible classical varieties there is a nonempty open subset $U_i\subseteq Z_i$ with $U_i\subseteq (Z_i{)}_{\mathrm{r}\mathrm{e}\mathrm{g}}\cap f^{-1}((W_i{)}_{\mathrm{r}\mathrm{e}\mathrm{g}})$ and with $d_x(f{\mathrm{\mid }}_{Z_i})\text{\hspace{0.17em}⁣}:T_x{Z}_i\to T_{f(x)}{W}_i$ surjective at every closed point $x\in U_i$; in particular $f(x)\in (W_i{)}_{\mathrm{r}\mathrm{e}\mathrm{g}}$. Choose a point $x\in U_i$, which is possible because $U_i$ is nonempty [F30], and put $y=f(x)\in (W_i{)}_{\mathrm{r}\mathrm{e}\mathrm{g}}$.

Closed subvarieties have injective differentials. Let $j\text{\hspace{0.17em}⁣}:Z^{\mathrm{\prime }}\hookrightarrow U^{\mathrm{\prime }}$ be the inclusion of a closed subvariety of an affine chart $U^{\mathrm{\prime }}$ of $X$, with vanishing ideal $I\subseteq B^{\mathrm{\prime }}=k[U^{\mathrm{\prime }}]$, so that $k[Z^{\mathrm{\prime }}]=B^{\mathrm{\prime }}\mathrm{/}I$ [F9, F22], and let $z\in Z^{\mathrm{\prime }}$ be a point. Then ${\mathcal{O}}_{Z^{\mathrm{\prime }},z}\cong (B^{\mathrm{\prime }}\mathrm{/}I{)}_{{\mathfrak{m}}_z}\cong B_{{\mathfrak{m}}_z}^{\mathrm{\prime }}\mathrm{/}I{B}_{{\mathfrak{m}}_z}^{\mathrm{\prime }}={\mathcal{O}}_{U^{\mathrm{\prime }},z}\mathrm{/}I{\mathcal{O}}_{U^{\mathrm{\prime }},z}$ by [F20] and [F21], the maximal ideal of this quotient is ${\mathfrak{m}}_z\mathrm{/}I{\mathcal{O}}_{U^{\mathrm{\prime }},z}$ with $I\subseteq {\mathfrak{m}}_z$, and $({\mathfrak{m}}_z\mathrm{/}I\mathcal{O})\mathrm{/}({\mathfrak{m}}_z\mathrm{/}I\mathcal{O}{)}^2\cong {\mathfrak{m}}_z\mathrm{/}({\mathfrak{m}}_z^2+I{\mathcal{O}}_{U^{\mathrm{\prime }},z})$ by [F19]; hence the natural map ${\mathfrak{m}}_z\mathrm{/}{\mathfrak{m}}_z^2\to {\mathfrak{m}}_{Z^{\mathrm{\prime }}}\mathrm{/}{\mathfrak{m}}_{Z^{\mathrm{\prime }}}^2$ is surjective [F19], and its dual is injective [F18]. By [F10] that dual is exactly the differential $d_{z}j\text{\hspace{0.17em}⁣}:T_z{Z}^{\mathrm{\prime }}\to T_z{U}^{\mathrm{\prime }}$ of the inclusion, so $d_{z}j$ is injective.

Determinantal description and local closedness. Let $x\in U$ be a closed point. Since the entries $a_{ij}(x)\in k$ are the images of the regular functions $a_{ij}$ under $B\to B\mathrm{/}{\mathfrak{m}}_x=k$, the $(r+1)\times (r+1)$ minors of $A(x)$ are the images of the corresponding minors of $A$; by [F18] the inequality $\mathrm{rank}A(x)\le r$ holds if and only if every one of those minors vanishes. By step 1.5 this says $\mathrm{rank}{d}_{x}f\le r$ if and only if every $(r+1)\times (r+1)$ minor of $A$ lies in ${\mathfrak{m}}_x$. Let $J_U\subseteq B$ be the ideal generated by all these minors; by [F22] the closed subvariety $V(J_U)\subseteq U$ has as its points exactly the maximal ideals of $B$ containing $J_U$, which are precisely the closed points $x\in U$ with $\mathrm{rank}{d}_{x}f\le r$. Hence $C_r\cap U=V(J_U)$ is closed in $U$.

Rank comparison. Let $U$ and $V$ be the affine charts around $x$ and $f(x)$ with $f(U)\subseteq V$ constructed in step 1.2; note $y=f(x)\in V$. The restrictions $Z_i\cap U\subseteq U$ and $W_i\cap V\subseteq V$ are closed subvarieties of these affine charts [F5] with inclusion differentials that, under the open-immersion tangent isomorphisms $Z_i\cap U\subseteq Z_i$, $U\subseteq X$, $W_i\cap V\subseteq W_i$ and $V\subseteq Y$ [F10], are the differentials $d_x\iota$ and $d_y\kappa$ of the closed inclusions $\iota \text{\hspace{0.17em}⁣}:Z_i\hookrightarrow X$ and $\kappa \text{\hspace{0.17em}⁣}:W_i\hookrightarrow Y$; by step 1.8 both $d_x\iota$ and $d_y\kappa$ are injective. The chain rule of [F10] applied to the identity $\kappa \circ (f{\mathrm{\mid }}_{Z_i})=f\circ \iota$ gives $d_y\kappa \circ d_x(f{\mathrm{\mid }}_{Z_i})=d_{x}f\circ d_x\iota$; since $d_y\kappa$ is injective, [F18] yields $\mathrm{rank}{d}_x(f{\mathrm{\mid }}_{Z_i})=\mathrm{rank}(d_{x}f\circ d_x\iota )\le \mathrm{rank}{d}_{x}f$. As $x\in Z_i\subseteq C_r$ we have $\mathrm{rank}{d}_{x}f\le r$, so $\mathrm{rank}{d}_x(f{\mathrm{\mid }}_{Z_i})\le r$.

Global closedness. The charts $U$ produced by step 1.2, as $x_0$ ranges over the classical points of $X$, cover $X$; by step 2.1 each $C_r\cap U$ is closed in $U$, hence $X\setminus C_r$ is open: a point $x\mathrm{\notin }{C}_r$ lies in one of these charts $U$, and $U\setminus (C_r\cap U)$ is an open neighbourhood of $x$ contained in $X\setminus C_r$. By [F26] the set $C_r$ therefore equals its closure in $X$ and is a closed subset of the classical variety $X$; with the reduced structure induced from $X$ it is a closed subvariety of $X$ [F5], which is claim 1.

Dimension of the image of each component. By step 1.7 the differential $d_x(f{\mathrm{\mid }}_{Z_i})$ is surjective, so by [F18] $\mathrm{rank}{d}_x(f{\mathrm{\mid }}_{Z_i})={\dim }_k{T}_y{W}_i\mathrm{.}$ Since $y\in (W_i{)}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ and $W_i$ is irreducible, [F27] gives ${\dim }_k{T}_y{W}_i={\dim }_y{W}_i$, and [F23] gives ${\dim }_y{W}_i=\dim W_i$. Therefore $\dim W_i=\mathrm{rank}{d}_x(f{\mathrm{\mid }}_{Z_i})\le r$ by step 2.2.

Assembling the closure. Since $C_r=Z_1\cup \cdots \cup Z_s$, one has $f(C_r)=f(Z_1)\cup \cdots \cup f(Z_s)$. The finite union $W_1\cup \cdots \cup W_s$ of the closed sets $W_i=\overline{f(Z_i)}$ is closed and contains $f(C_r)$, so $\overline{f(C_r)}\subseteq W_1\cup \cdots \cup W_s$ by [F26]; conversely each $W_i=\overline{f(Z_i)}$ is contained in $\overline{f(C_r)}$ because $f(Z_i)\subseteq f(C_r)$ and $\overline{f(C_r)}$ is closed [F26]. Hence $\overline{f(C_r)}=W_1\cup \cdots \cup W_s$, a finite union of closed subsets of the Noetherian space $Y$ [F6]; by [F25] its dimension is ${\max }_i\dim W_i$, which is at most $r$ by step 3.2, and for $C_r=\mathrm{\varnothing }$, when the list $W_1,\dots ,W_s$ is empty, both the maximum and $\dim \mathrm{\varnothing }$ are $-\mathrm{\infty }$ by [F24], which is at most $r$. This is claim 2.

Boundary and scope dispositions. Empty: $X$ and $Y$ may be empty or reducible, and no irreducibility is assumed; for $X=\mathrm{\varnothing }$ the set $C_r$ is empty and closed and $\dim \overline{f(C_r)}=\dim \mathrm{\varnothing }=-\mathrm{\infty }\le r$ by [F24], while for $Y=\mathrm{\varnothing }$ also $X=\mathrm{\varnothing }$ because a morphism into the empty scheme has empty source. Zero: $r=0$ is allowed, and then $C_0$ consists of the points where the differential vanishes; the relative dimension $n-c=0$ of step 1.2 is allowed, in which case ${\mathrm{\Omega }}_{B\mathrm{/}k}=0$ by [F14], the matrix $A$ has no rows, its rank is $0$, and $C_r\cap U=U$ for every $r\ge 0$ with $\dim \overline{f(U)}\le \dim V=0$ when $V$ is a point. One: nothing in the argument divides by a natural number or requires a positive relative dimension or a positive number of coordinate functions; for $m=0$, when $V$ is a single point, the matrix $A$ has no columns, $\gamma =0$ and $C_r\cap U=U$, in agreement with claim 2. Degenerate: the smoothness of $X$ in claim 1 is essential and cannot be dropped — for the singular closed subvariety $X=V(xy)\subseteq {\mathbf{A}}_k^2$, the morphism $f\text{\hspace{0.17em}⁣}:X\to {\mathbf{A}}_k^1$, $(x,y)\mapsto x$, has $\mathrm{rank}{d}_{(0,b)}f=0$ for every $b\mathrm{\ne }0$ while $\mathrm{rank}{d}_{(0,0)}f=1$, so the corresponding $C_0$ is the punctured $y$-axis, not closed; reducible and disconnected smooth $X$ are nevertheless allowed, and claim 2 does not use the smoothness of $Y$, the argument for it needing only the local presentation of $Y$ with coordinate functions generating its coordinate ring. Endpoints: the integer $r$ ranges over $r\ge 0$ with no upper bound, and claim 2 is trivial for $r\ge \dim Y$ since $\dim \overline{f(C_r)}\le \dim Y$; the marginal case $r=0$ with $C_0=\mathrm{\varnothing }$ is covered by the empty-list convention $-\mathrm{\infty }\le 0$ of step 4.1, while the case in which $U_i$ is a single point is covered because the surjectivity conclusion of [F28] persists at every closed point of $U_i$. Nonempty-choice: AC is declared as [F1] and is used exactly through the AC-assuming suppliers [F4] (classical-scheme dictionary), [F6] (finite component decompositions), [F7] (principal opens), [F20] (local rings), [F22] (Nullstellensatz correspondence), [F27] (regular-point tangent criterion), [F28] (generic differential surjectivity) and [F31] (closures of irreducible sets), cited at steps 1.1, 1.2, 1.6, 1.7, 1.8 and 4.1; the chart, matrix, determinantal and rank-comparison computations of steps 1.3, 1.4, 1.5, 2.1, 2.2 and the local-ring duality of step 1.8 are choice-free, and no family of nonempty sets is selected anywhere. Biconditional directions: the statement asserts no equivalence, so the forward and reverse directions of a biconditional are not applicable; the only equivalence used inside the proof is the determinantal criterion of [F18], applied in step 2.1 in the direction "all $(r+1)$-minors vanish implies rank at most $r$" and conversely, and the cotangent isomorphisms of [F12] are used only through their naturality.