Generic smoothness over a dense target open

Setup and conventions. By [F2] the classical varieties $X$ and $Y$ are quasi-compact locally ringed spaces over $k$ covered by affine models, every point of either is a closed point with residue field $k$, polynomial principal opens form a basis of the topology, and their structure sheaves are sheaves of $k$-valued functions; by [F27] the irreducible classical varieties $X$ and $Y$ correspond to integral, hence reduced, finite-type $k$-schemes and $f$ to a $k$-morphism of those schemes; they are Noetherian by [F6]. Write $m=\dim X$ and $n=\dim Y$ [F7]. The field $k$ is perfect by [F13], and by hypothesis the structure morphism $X\to \mathrm{Spec}k$ is smooth [F3], so [F14] makes $X$ regular. Every $k$-morphism of finite-type $k$-schemes is of finite type: on affine charts $\mathrm{Spec}A\subseteq Y$ and $\mathrm{Spec}B\subseteq X$ with $f(\mathrm{Spec}B)\subseteq \mathrm{Spec}A$, the algebra $B$ is generated as an $A$-algebra by finitely many $k$-algebra generators [F4], so $f$ is locally of finite type [F5], and it is quasi-compact because $X$ is Noetherian, so that every open subset of $X$ is quasi-compact [F5, F6]; the same argument applies to the restriction of $f$ to any open subvariety of $X$.

The non-dominant case. Suppose $f$ is not dominant, so the closure $Z:=\overline{f(X)}$ is a closed subset of $Y$ with $Z\mathrm{\ne }Y$ [F10]. Its complement $U:=Y\setminus Z$ is open and nonempty, and it is dense in $Y$ because a nonempty open subset of the irreducible space $Y$ is dense [F9]; moreover $f(X)\subseteq Z$, so $f^{-1}(U)=\mathrm{\varnothing }$. The empty morphism $\mathrm{\varnothing }\to U$ is smooth by [F3], the standard-smooth condition being imposed at every source point and the empty source having none; the empty scheme is a classical variety and the morphism is of finite type because its source is quasi-compact. Thus claim 1 holds in this case with this $U$.

The dominant case: reduction to the regular locus of the target. Suppose now that $f$ is dominant. The regular locus $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ [F12] of $Y$, a reduced finite-type $k$-scheme over the perfect field $k$ [F27], is a nonempty open subset of $Y$ whose intersection with every irreducible component of $Y$ is dense open in that component [F11]; since $Y$ is irreducible, $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is nonempty, open and dense, hence irreducible [F9], and $\dim Y_{\mathrm{r}\mathrm{e}\mathrm{g}}=\dim Y=n$ [F8]. At every $y\in Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ the local ring ${\mathcal{O}}_{Y_{\mathrm{r}\mathrm{e}\mathrm{g}},y}\cong {\mathcal{O}}_{Y,y}$ is regular [F12, F15, F16], and $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is of finite type over the perfect field $k$ [F2, F13], so $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}\to \mathrm{Spec}k$ is smooth by [F14]. The open subvariety $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is itself a classical variety over $k$: it is quasi-compact because $Y$ is Noetherian [F6], and its intersections with the affine models of $Y$ are covered by principal opens, which are affine models by [F17]. Similarly $X_0:=f^{-1}(Y_{\mathrm{r}\mathrm{e}\mathrm{g}})$ is an open subvariety of $X$, hence a classical variety over $k$, it is nonempty because the dense subset $f(X)$ meets the nonempty open set $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ [F9, F10], it is irreducible with $\dim X_0=\dim X=m$ [F8, F9], and its structure morphism $X_0\to \mathrm{Spec}k$ is smooth by locality on the source [F3]. The restriction $f_0:=f{\mathrm{\mid }}_{X_0}\text{\hspace{0.17em}⁣}:X_0\to Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is a finite-type morphism of classical varieties: on affine charts $\mathrm{Spec}A\subseteq Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ and $\mathrm{Spec}B\subseteq X_0$ with $f_0(\mathrm{Spec}B)\subseteq \mathrm{Spec}A$ the algebra $B$ is generated as an $A$-algebra by finitely many $k$-algebra generators [F4, F5], and $f_0$ is quasi-compact because $X_0$ is Noetherian, so that every open subset of $X_0$ is quasi-compact [F5, F6].

The rank-$(n-1)$ locus and the open set. Let $C=\{x\in X_0:\mathrm{rank}{d}_x{f}_0\le n-1\}$. If $n\ge 1$, then [F18] applied to the morphism $f_0$ of smooth classical varieties with $r=n-1\ge 0$ shows that $C$ is closed in $X_0$ and that the closed subvariety $Z:=\overline{f_0(C)}\subseteq Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ satisfies $\dim Z\le n-1$; if $n=0$, then $C=\mathrm{\varnothing }$ because ranks are nonnegative, so $Z=\mathrm{\varnothing }$ and $\dim Z=-\mathrm{\infty }\le -1$ [F19]. In either case $Z\mathrm{\ne }{Y}_{\mathrm{r}\mathrm{e}\mathrm{g}}$: when $n\ge 1$ because $\dim Y_{\mathrm{r}\mathrm{e}\mathrm{g}}=n>n-1\ge \dim Z$ by step 1.3, and when $n=0$ because $Z=\mathrm{\varnothing }$ while $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}\mathrm{\ne }\mathrm{\varnothing }$ by step 1.3. Put $U:=Y_{\mathrm{r}\mathrm{e}\mathrm{g}}\setminus Z$; then $U$ is open in $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ and in $Y$, it is nonempty because $Z\mathrm{\ne }{Y}_{\mathrm{r}\mathrm{e}\mathrm{g}}$, and it is dense in $Y$ because a nonempty open subset of the irreducible space $Y$ is dense [F9]. Also $X^{\mathrm{\prime }}:=f^{-1}(U)=f_0^{-1}(U)$ is a nonempty open subvariety of $X_0$ because $f$ is dominant and $U$ is nonempty open [F9, F10].

The rank equals $n$ on $f^{-1}(U)$. Let $x$ be a classical closed point of $X^{\mathrm{\prime }}=f^{-1}(U)$, so $x\in X_0$ and $f(x)\in U\subseteq Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ by step 2.1. Then $x\mathrm{\notin }C$, so $\mathrm{rank}{d}_x{f}_0\ge n$ by the definition of $C$; on the other hand $\mathrm{rank}{d}_x{f}_0\le {\dim }_k{T}_{f(x)}{Y}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ because $d_x{f}_0$ is a $k$-linear map into that finite-dimensional space. Since $f(x)$ is a regular point of the classical variety $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ we have ${\dim }_k{T}_{f(x)}{Y}_{\mathrm{r}\mathrm{e}\mathrm{g}}={\dim }_{f(x)}{Y}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ [F12], and since $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is irreducible of dimension $n$ [step 1.3] this equals $\dim Y_{\mathrm{r}\mathrm{e}\mathrm{g}}=n=\dim Y$ [F7, F8]. Hence $\mathrm{rank}{d}_x{f}_0=n$, and $d_x{f}_0$ is surjective.

From closed points to every scheme point. At each classical closed point $x\in X^{\mathrm{\prime }}$ the morphism $f_0:X_0\to Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is between smooth classical varieties with their smooth scheme structures and is of finite type by step 1.3. Its differential is surjective by step 3.1, so [F20] gives smoothness at $x$. Restricting over $U$ preserves this local property by [F3]; hence $g:X^{\mathrm{\prime }}=f^{-1}(U)\to U$ is smooth at every closed point. Let $S\subseteq X^{\mathrm{\prime }}$ be the scheme smooth locus of $g$. It is open: a standard smooth presentation after principal shrinking, as in [F3], witnesses smoothness at every prime of that shrinking, since its Jacobian minor is a unit there. If $X^{\mathrm{\prime }}\setminus S$ were nonempty, intersect it with an affine chart $\mathrm{Spec}B$ of the finite-type scheme $X^{\mathrm{\prime }}$. The intersection is a nonempty closed subset, and [F28] gives a closed point of that chart in it. By [F27] this is a classical point of $X^{\mathrm{\prime }}$, contrary to the closed-point conclusion just proved. Thus $S=X^{\mathrm{\prime }}$, and $g$ is smooth at every scheme point. This proves claim 1.

The fibres over the further open set. Suppose $f$ is dominant and let $U$ be the dense open set of step 2.1, over which $f^{-1}(U)\to U$ is smooth by step 4.1. By [F26] there is a nonempty open $V_1\subseteq Y$, contained in $f(X)$, such that for every closed point $y\in V_1$ the fibre $f^{-1}(y)$ is nonempty and of pure dimension $r=m-n=\dim X-\dim Y$; put $V:=U\cap V_1$, a nonempty open subset of the irreducible $Y$, hence dense [F9]. For a closed point $y\in V$, so that $k(y)=k$, the scheme-theoretic fibre $X_y=X{\times }_Y\mathrm{Spec}k(y)$ [F21] has underlying topological space $f^{-1}(y)$ by [F22], so $X_y$ is nonempty. The classical fibre in [F26] is its closed-point space. Closed-point density [F28] identifies closed subsets and irreducible components of the scheme fibre with their classical traces, chart by chart, preserving strict chains and dimensions; nilpotents do not affect these spaces. Thus $X_y$ has pure dimension $r$. For smoothness, the fibre product $X{\times }_{Y}U$ is represented by the open subscheme $f^{-1}(U)\subseteq X$ by [F23], and the universal property of fibre products [F24] gives a canonical isomorphism $X_y\cong f^{-1}(U){\times }_U\mathrm{Spec}k(y)$; the projection on the right is the base change of the smooth morphism $f^{-1}(U)\to U$ along $\mathrm{Spec}k(y)\to U$, hence is smooth over $k$ because locally standard smooth morphisms are stable under base change [F25]. Therefore every fibre $X_y$ with closed $y\in V$ is nonempty, smooth over $k$, and of pure dimension $r=\dim X-\dim Y$.

Boundary and scope dispositions. Empty: in the non-dominant case $f^{-1}(U)=\mathrm{\varnothing }$ and the empty morphism $\mathrm{\varnothing }\to U$ is smooth vacuously (step 1.2); in the dominant case $X$ and $Y$ are nonempty because irreducible [F9], the open sets $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$, $U$ and $V$ are nonempty by steps 1.3, 2.1 and 5.1, and the fibres over $V$ are nonempty by step 5.1, so no empty-fibre convention is invoked in claim 2. Zero: the target dimension $n=0$ is admitted; then $C=Z=\mathrm{\varnothing }$, $U=Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ and the rank computation of step 3.1 reads $0\le \mathrm{rank}{d}_x{f}_0\le 0$, while the fibre clause gives a single fibre of pure dimension $r=m$; the relative dimension $r=0$ is likewise admitted in step 5.1, where smooth fibres of pure dimension zero are finite reduced $k$-schemes, and nothing in the argument divides by $r$. One: no step divides by a natural number, selects a basis, or requires a positive dimension, codimension, or number of equations; the cases $n=1$ with $r=0$ or $r=1$ are covered by the same steps 2.1 through 5.1. Degenerate: the smoothness of $X$ is essential for claim 1 and is used through the critical-locus bound [F18] in step 2.1 and the submersion criterion [F20] in step 4.1; the constant cusp family $X=\mathrm{Spec}k[x,y,z]\mathrm{/}(y^2-x^3)\to {\mathbb{A}}_k^1$, $(x,y,z)\mapsto z$, is a dominant morphism of irreducible classical varieties over $k$ to a smooth target whose every fibre is the singular cusp, so that no nonempty open $U$ has $f^{-1}(U)\to U$ smooth, as recorded on the counterexample page of this pair. The target $Y$ need not be smooth outside $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$: for the fold $f\text{\hspace{0.17em}⁣}:{\mathbb{A}}_k^1\to {\mathbb{A}}_k^1$, $t\mapsto t^2$, the differential vanishes at $0$, the fibre over $0$ is the non-reduced $\mathrm{Spec}k[ϵ]\mathrm{/}({ϵ}^2)$, and $U={\mathbb{A}}_k^1\setminus \{0\}$ is the best possible dense open, while the constant morphism ${\mathbb{A}}_k^1\to {\mathbb{A}}_k^1$ with value $0$ has $f^{-1}(U)=\mathrm{\varnothing }$ for $U={\mathbb{A}}_k^1\setminus \{0\}$; the fibres of step 5.1 are not asserted to be irreducible or connected. Endpoints: the statement has no interval parameter; the boundary $n=0$ versus $n\ge 1$ is handled in step 2.1 through the convention $\dim \mathrm{\varnothing }=-\mathrm{\infty }\le -1$ of [F19] and the trivial lower bound in step 3.1, and the open sets $U,V$ are dense but need not be all of $Y$, as the fold example shows; the generic fibre dimension is constant over $V$ by construction and not merely bounded. Nonempty-choice: AC is declared as [F1] and enters exactly through the AC-assuming suppliers [F11] (density of the regular locus), [F12] (the classical regular-point tangent test), [F14] (regularity versus smoothness), [F18] (the critical-locus dimension bound), [F20] (the submersion criterion) [F26] (generic fibre dimension), and [F28] (closed-point density), cited at steps 1.3, 2.1, 3.1, 4.1 and 5.1; the finite-type verification of step 1.1, the fibre-product pasting and base-change smoothing of step 5.1, and the remaining linear algebra 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 two equivalences used in the proof — the submersion criterion [F20], applied in the direction "surjective differential at a classical point implies smoothness there" in step 4.1, and regularity-versus-smoothness [F14], applied in the direction "regular implies smooth" in step 1.3 for $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ — are used only in those directions, and no converse of the theorem is claimed.