General hypersurfaces give smooth complete intersections

Setup and the dictionary. By [F1] AC is available. The field $k$ is algebraically closed of characteristic $0$, hence perfect and infinite by [F3] and [F11]. By [F2] the variety $X$ is a nonempty irreducible projective algebraic set of pure dimension $d$, so it is reduced by the definition in [F6]; under the equivalence [F7] the associated finite-type $k$-scheme $X$ is integral and separated, with closed points corresponding to classical points and with residue field $k$ at every closed point. The structure morphism $X\to \mathrm{Spec}k$ is smooth in the sense of [F9], so by [F12] every local ring of $X$ is regular, and since $X$ is irreducible with component $X$ of dimension $d$, [F16] gives $\dim {\mathcal{O}}_{X,x}=d$ for every closed point $x\in X$; by [F9] the nonempty open subvarieties $X_j=X\cap D_{+}(x_j)$ are smooth over $k$ of pure dimension $d$, and by [F19] they are irreducible of dimension $d$.

The parameter spaces. For $e\ge 1$ put $S_e=k[x_0,\dots ,x_N{]}_e$, a finite-dimensional $k$-vector space by [F4], and let $\mathbf{P}(S_e)=(S_e\setminus \{0\})\mathrm{/}{k}^{\times }$ be the space of lines with the projective Zariski topology of [F23]; choosing a basis identifies it with a projective space ${\mathbf{P}}_k^{M_e}$, whose dimension is $M_e$. Its homogeneous vanishing ideal is $(0)$: if a nonzero homogeneous polynomial vanished at every point of ${\mathbf{P}}_k^{M_e}$, it would vanish at every nonzero tuple of $k^{M_e+1}$ and also at the zero tuple when its degree is positive, contradicting the polynomial nonvanishing theorem [F41] over the infinite field $k$; a nonzero constant cannot vanish anywhere. The zero ideal is prime because the coordinate polynomial ring over $k$ is a domain [F4], so $\mathbf{P}(S_e)$ is a nonempty irreducible projective variety by [F2] and [F24]. Consequently ${\mathrm{\Pi }}_r=\mathbf{P}(S_{e_1}){\times }_k\cdots {\times }_k\mathbf{P}(S_{e_r})$ is a nonempty classical projective variety for every $r\ge 1$, by [F30] applied iteratively, and ${\mathrm{\Pi }}_0=\mathrm{Spec}k$ is a one-point classical variety; the classical points of $\mathbf{P}(S_e)$ are exactly the lines $[F]$ with $0\mathrm{\ne }F\in S_e$ by [F23], and those of ${\mathrm{\Pi }}_r$ are the tuples of lines.

The intersection subschemes and their local equations. For $0\mathrm{\ne }F\in S_e$ let $V_{+}(F)$ be the zero subscheme of the section of ${\mathcal{O}}_{{\mathbf{P}}_k^N}(e)$ represented by $F$, supplied by [F40]; on the standard chart $D_{+}(x_j)$ is cut out by the dehomogenized form $f=(F\mathrm{/}{x}_j^e)$ regarded as a polynomial in the coordinates $y_1,\dots ,y_N$ of [F5]; the two dehomogenizations of $F$ on an overlap $D_{+}(x_j)\cap D_{+}(x_k)$ differ by the unit $(x_k\mathrm{/}{x}_j{)}^e$ (with $x_j=1$ on the first chart), so the principal ideals agree and [F40] gives the well-defined closed subscheme with underlying set the classical hypersurface $V_{+}(F)$ of [F3]; clearly $(F)=(\lambda F)$ for $\lambda \mathrm{\ne }0$ by [F4], so $V_{+}(F)$ depends only on the line $[F]$. For a tuple $F_1,\dots ,F_r$ put $Z(F_1,\dots ,F_r)=X\cap V_{+}(F_1)\cap \cdots \cap V_{+}(F_r)$, the scheme-theoretic intersection of closed subschemes of ${\mathbf{P}}_k^N$ in the sense of [F21], a closed subscheme of $X$ depending only on the parameter point of [F23] in ${\mathrm{\Pi }}_r$; for $r=0$ the intersection is $X$ by the empty-family clause of [F21]. On the chart $D_{+}(x_j)$ one has $X\cap D_{+}(x_j)=X_j=\mathrm{Spec}{A}_j$ with $A_j=k[y_1,\dots ,y_N]\mathrm{/}{I}_j$ a finite-type $k$-algebra [F39], and by [F21] and [F22] the restriction of $Z(F_1,\dots ,F_r)$ to $D_{+}(x_j)$ is the closed subscheme $\mathrm{Spec}(A_j\mathrm{/}(f_1,\dots ,f_r))$, where $f_i$ is the dehomogenization of $F_i$.

Closed points of the intersections. Let $Z\subseteq {\mathbf{P}}_k^N$ be any closed subscheme of finite type over $k$, for instance $Z(F_1,\dots ,F_r)$, with the induced reduced projective algebraic set as its underlying space. A point $x\in Z$ is a closed point of $Z$ if and only if $\kappa (x)=k$: closedness of the singleton is local on the finite affine chart cover $Z\cap D_{+}(x_j)=\mathrm{Spec}{B}_j$ with $B_j$ finite type over $k$ [F5, F39], and on an affine finite-type $k$-algebra a prime is maximal if and only if its residue field is finite over $k$, hence equal to $k$ because $k$ is algebraically closed [F8]; moreover every nonempty open subset of $Z$ contains a closed point of $Z$, because it meets some chart and [F8] supplies a closed point of that chart's spectrum, which has residue field $k$ and is closed in $Z$ by the first assertion.

The case of no forms. If $r=0$ then ${\mathrm{\Pi }}_0=\mathrm{Spec}k$ by 1.2, the intersection is $Z=X$ by 1.3, and $X$ is nonempty, smooth over $k$ and of pure dimension $d$ by hypothesis and [F2]; so $U={\mathrm{\Pi }}_0$ exhibits claim 1 in this case.

The Veronese transfer of Bertini. Let $Y\subseteq {\mathbf{P}}_k^N$ be a nonempty closed subscheme which is smooth over $k$ of pure dimension $m\ge 1$ (for instance an intersection produced below), let $e\ge 1$, and consider the degree-$e$ Veronese map $\nu ={\nu }_{e,N}\text{\hspace{0.17em}⁣}:{\mathbf{P}}_k^N\to {\mathbf{P}}_k^M$ of [F28]; the composite of the closed immersions $Y\hookrightarrow {\mathbf{P}}_k^N\stackrel{\nu }{\to }{\mathbf{P}}_k^M$ is again a closed immersion by [F40]; write $Y^{\mathrm{\prime }}$ for its closed scheme image, which is isomorphic to $Y$ by that composite, so $Y^{\mathrm{\prime }}$ is nonempty, smooth over $k$ of pure dimension $m$, and it is the image of a closed immersion into ${\mathbf{P}}_k^M$. Applying the "in particular" clause of [F29] to the closed immersion $Y^{\mathrm{\prime }}\hookrightarrow {\mathbf{P}}_k^M$ produces a nonempty Zariski-open subset $U\subseteq \mathbf{P}(W_h)$ of the hyperplane parameter space of the embedding, such that for every closed point $[s]\in U$ and every degree-one form $L$ with $L{\mathrm{\mid }}_{Y^{\mathrm{\prime }}}=s$, the scheme-theoretic hyperplane section $Y^{\mathrm{\prime }}{\times }_{{\mathbf{P}}_k^M}{V}_{+}(L)$ is smooth over $k$. The coefficient assignment $L\mapsto L\circ \nu$ is a linear isomorphism from the space of linear forms on ${\mathbf{P}}_k^M$ onto $S_e$ (both are $k$-vector spaces with basis indexed by the degree-$e$ monomials), and by [F27] (applicable with $n=N\ge 1$, since $m\ge 1$) the associated linear form $L_F$ of $0\mathrm{\ne }F\in S_e$ satisfies $L_F(\nu ([x]))=F(x)$ and ${\nu }^{-1}(V_{+}(L_F))=V_{+}(F)$; comparing dehomogenized equations on the standard charts as in 1.3, the local equations identify $Y\cap V_{+}(F)$ with the fibre product $Y^{\mathrm{\prime }}{\times }_{{\mathbf{P}}_k^M}{V}_{+}(L_F)$ over the isomorphism $Y\to Y^{\mathrm{\prime }}$ from [F40]; let $K\subseteq S_e$ be the subspace of forms restricting to zero on $Y^{\mathrm{\prime }}$, the kernel of the $k$-linear map $F\mapsto L_F{\mathrm{\mid }}_{Y^{\mathrm{\prime }}}$, which is proper because $Y\mathrm{\ne }\mathrm{\varnothing }$, and let $\pi \text{\hspace{0.17em}⁣}:\mathbf{P}(S_e)\setminus \mathbf{P}(K)\to \mathbf{P}(W_h)$ be the induced morphism. Then $\pi$ is defined on a nonempty open subset, it carries $k$-rational points to $k$-rational points because it is induced by a $k$-linear map, and it is surjective because the composite $S_e\to \mathrm{\Gamma }({\mathbf{P}}_k^M,\mathcal{O}(1))\to W_h$ is onto by definition of the hyperplane system $W_h$ as the span of the restricted coordinate forms. Hence $V:={\pi }^{-1}(U)$ is a nonempty open subset of $\mathbf{P}(S_e)$, and every closed point $[F]\in V$ is good: by the criterion of 1.4 the point $[F]$ has residue field $k$, hence so does its image $\pi ([F])$, so $\pi ([F])$ is a closed point of $\mathbf{P}(W_h)$ lying in $U$, and the identification above together with [F29] makes $Y\cap V_{+}(F)\cong Y^{\mathrm{\prime }}{\times }_{{\mathbf{P}}_k^M}{V}_{+}(L_F)$ smooth over $k$.

The universal intersection and the nonemptiness locus. In the universal-intersection and defect-locus constructions all parameter-space and incidence loci are classical loci of $k$-points; [F31] is applied only in that category. The open-set correspondence of [F7] on the irreducible parameter variety ${\mathrm{\Pi }}_r$ gives a scheme open with exactly the same closed points for each classical open constructed here. Every fibre assertion in the remainder of the proof is for a closed parameter $t$, so $\kappa (t)=k$. For $1\le i\le r$ let $W_i\subseteq {\mathbf{P}}_k^N{\times }_k{\mathrm{\Pi }}_r$ be the set of pairs $(x,[F_1],\dots ,[F_r])$ with $F_i(x)=0$; on a product of a standard chart of ${\mathbf{P}}_k^N$ [F5] and affine charts of the factors $\mathbf{P}(S_{e_i})$ (normalizing one coefficient of each form to $1$), the condition is cut out by the polynomial obtained by dehomogenizing $F_i$, so $W_i$ is closed and so is the intersection $W_r={\bigcap }_i{W}_i$. Put $\mathcal{W}:=W_r\cap (X{\times }_k{\mathrm{\Pi }}_r)$, a closed subset of $X{\times }_k{\mathrm{\Pi }}_r$: its fibre over a classical parameter $t\in {\mathrm{\Pi }}_r$ is exactly the classical closed-point set of $Z(t)=X\cap V_{+}(F_1)\cap \cdots \cap V_{+}(F_r)$ by [F21] and the local description of 1.3. The projection $q\text{\hspace{0.17em}⁣}:X{\times }_k{\mathrm{\Pi }}_r\to {\mathrm{\Pi }}_r$ is a closed map: $X$ is closed in ${\mathbf{P}}_k^N$ and nonempty [F2], ${\mathrm{\Pi }}_r$ is a classical variety [F30], so by [F31] the projection ${\mathrm{\Pi }}_r{\times }_k{\mathbf{P}}_k^N\to {\mathrm{\Pi }}_r$ is closed, and a closed subset of the closed subset $X{\times }_k{\mathrm{\Pi }}_r$ has closed image under its restriction; since $\mathcal{W}\subseteq X{\times }_k{\mathrm{\Pi }}_r$ is closed in ${\mathbf{P}}_k^N{\times }_k{\mathrm{\Pi }}_r$, the image $N_r:=q(\mathcal{W})=\{t\in {\mathrm{\Pi }}_r:Z(t)\mathrm{\ne }\mathrm{\varnothing }\}$ is closed, and its complement $\{t:Z(t)=\mathrm{\varnothing }\}$ is open. Moreover $N_r={\mathrm{\Pi }}_r$ whenever $0\le r\le d$. For $r=0$ this is $Z=X\mathrm{\ne }\mathrm{\varnothing }$ by the hypothesis on $X$. Given any tuple of $r\ge 1$ forms, begin with the irreducible closed set $C_0=X$ of dimension $d$. Inductively, if $i\le r\le d$ and an irreducible closed set $C_{i-1}\subseteq Z(F_1,\dots ,F_{i-1})$ has dimension $m\ge d-i+1\ge 1$, then either $F_i$ vanishes identically on $C_{i-1}$, in which case take $C_i=C_{i-1}$, or [F26] gives a nonempty irreducible component $C_i$ of $C_{i-1}\cap V_{+}(F_i)$ of dimension $m-1$. In both cases $C_i\subseteq Z(F_1,\dots ,F_i)$ and $\dim C_i\ge d-i$. Thus the final intersection is nonempty for every closed parameter tuple. Thus $N_r={\mathrm{\Pi }}_r$ as classical loci for $r\le d$, which proves the required nonemptiness for every closed parameter. The emptiness locus for any $r$ is a classical open, hence corresponds to a scheme open by [F7]. [F2, F5, F8, F21, F26, F30, F31, F7, step 1.3, induction] 3.1 The dimension and nonemptiness step. Let $Y\subseteq {\mathbf{P}}_k^N$ and $e\ge 1$ be as in 2.2, with $m\ge 1$, and let $V={\pi }^{-1}(U)\subseteq \mathbf{P}(S_e)$ be the nonempty open set of 2.2, whose closed points are the forms $F$ with $Y\cap V_{+}(F)$ smooth over $k$. The irreducible components $Y_1,\dots ,Y_t$ of $Y$ are finite in number and closed by [F17], each is a nonempty closed subvariety of ${\mathbf{P}}_k^N$ of dimension $m$ [F2, F19] (pure dimension $m$ means every component has dimension $m$), and $F$ vanishes on $Y_l$ exactly when $F\in I(Y_l{)}_e$, a proper linear subspace of $S_e$: since $Y_l\mathrm{\ne }\mathrm{\varnothing }$, some coordinate function $x_m$ is nonzero at a point of $Y_l$, and then $x_m^e\mathrm{\notin }I(Y_l{)}_e$ [F3, F4]; thus the set $W$ of forms not vanishing on any component of $Y$ is the complement of finitely many proper closed subsets, hence open and nonempty by [F25]. Both $V$ and $W$ are nonempty open in the irreducible space $\mathbf{P}(S_e)$ of 1.2, so $V\cap W$ is nonempty and open by [F18], and by 1.4 (applied to the projective space $\mathbf{P}(S_e)$) it contains a closed point $[F]$; by 2.2 this $F$ satisfies that $Y\cap V_{+}(F)$ is smooth over $k$, and in particular $F\mathrm{\ne }0$. For each $l$, $F$ does not vanish on $Y_l$ and $\dim Y_l=m\ge 1$, so [F26] gives that $Y_l\cap V_{+}(F)$ is nonempty and has all components of dimension $m-1$; every component of the finite union $Y\cap V_{+}(F)={\bigcup }_l(Y_l\cap V_{+}(F))$ is contained in one of the closed pieces and contains a component of one of them, so by [F18] every component of $Y\cap V_{+}(F)$ has dimension exactly $m-1$.

The rank defect locus is closed, so the full-rank locus is open. For $0\le r\le d$, fix once and for all a finite generating list $g_1,\dots ,g_s$ of the ideal $I_j$ of $X_j$ in each chart $D_{+}(x_j)$, possible by [F39]. On the product of such a chart with affine charts of all factors of ${\mathrm{\Pi }}_r$, the dehomogenized forms $f_1,\dots ,f_r$ and the $g_l$ are polynomials, and we differentiate only in the $N$ ambient coordinates, holding parameter coefficients constant, to obtain the rows of the combined Jacobian matrix $J_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}$ of $g_1,\dots ,g_s,f_1,\dots ,f_r$; define $B_j\subseteq D_{+}(x_j){\times }_k{\mathrm{\Pi }}_r$ to be the common zero locus of all $g_l$, all $f_i$ and all $(N-d+r)\times (N-d+r)$ minors of $J_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}$. This locus is closed in the product chart, since all displayed functions are polynomial there. At every classical pair $(x,t)$, both residue fields equal $k$. By [F34] the module of differentials of the chart ring of this fixed $k$-fibre $Z(t)$ over $k$ is the cokernel of the transpose of $J_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}$, so its fibre dimension over $x$ equals $N-{\mathrm{rank}}_{\kappa (x)}{J}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}$ (the rank of a matrix is unchanged by transposition); this number depends only on the point $x$ and the tuple $t$, not on the chart or the chosen finite generating list, because $\mathrm{\Omega }$ and its base change do not. Hence the closed loci $B_j$ agree on overlaps and, closedness being local on an open cover, they glue to one closed subset $B\subseteq X{\times }_k{\mathrm{\Pi }}_r$: the locus of pairs $(x,t)$ with $x\in Z(t)$ and ${\dim }_{\kappa (x)}({\mathrm{\Omega }}_{Z(t)\mathrm{/}k}\otimes \kappa (x))>d-r$. By the classical closed projection of step 2.3, $q(B)$ is classically closed. Consequently its classical complement corresponds under [F7] to a scheme open $U_r^{\mathrm{\prime }}\subseteq {\mathrm{\Pi }}_r$, whose closed parameters are exactly those with no classical rank-defect pair. No assertion about ${\mathrm{\Omega }}_{Z(t)\mathrm{/}k}$ for nonclosed parameters is used.

Existence of a good tuple for $0\le r\le d$. We claim that for every $0\le i\le \min (r,d)$ there are forms $F_1,\dots ,F_i$, $0\mathrm{\ne }{F}_l\in S_{e_l}$, such that $Z_i=Z(F_1,\dots ,F_i)$ is nonempty, smooth over $k$ and of pure dimension $d-i$. For $i=0$ this is 1.1 and 2.1. For the induction step, let $1\le i\le \min (r,d)$, so that $Y=Z_{i-1}$ is a nonempty closed subscheme of ${\mathbf{P}}_k^N$ which is smooth over $k$ of pure dimension $d-i+1\ge 1$ and reduced (regular by [F12], hence a domain at each local ring by [F14]); applying 2.2 and 3.1 with $e=e_i$ and $m=d-i+1$ produces $F_i\in S_{e_i}$ such that $Z_i=Y\cap V_{+}(F_i)$ is nonempty, smooth over $k$ and of pure dimension $d-i$. In particular, for $0\le r\le d$ there is a tuple $t_0=(F_1,\dots ,F_r)$ with $Z(t_0)$ nonempty, smooth over $k$ and of pure dimension $d-r$.

Closed points of full-rank tuples are regular of dimension $d-r$. Let $t$ be a closed point of $U_r^{\mathrm{\prime }}$, let $x\in Z(t)$ be a closed point, and work in a chart $D_{+}(x_j)$ containing $x$ with the notation of 3.2. Since $t\mathrm{\notin }q(B)$, the rank of $J_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}(x)$ over $\kappa (x)=k$ (step 1.4) is at least $N-d+r$; on the other hand the $g$-block has rank $N-d$, because $X_j$ is smooth over $k$ hence regular at the rational point $x$ [F9, F12] and [F33] (rational-point clause together with the classical dimension clause, since $\dim {\mathcal{O}}_{X,x}=d$ by 1.1) gives $\mathrm{rank}J(g)(x)=N-\dim (A_j{)}_{{\mathfrak{m}}_x}=N-d$, where ${\mathfrak{m}}_x$ is the maximal ideal of $A_j$ corresponding to $x$, while the $f$-block adds at most its $r$ rows; hence $\mathrm{rank}{J}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}(x)=N-d+r$. By [F32] applied to the actual ideal of $Z(t)$ in the chart, whose finite generating list is $g_1,\dots ,g_s,f_1,\dots ,f_r$, we get ${\dim }_k{T}_{x}Z(t)=N-\mathrm{rank}{J}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}(x)=d-r$. Consider the morphism of classical varieties $g\text{\hspace{0.17em}⁣}:X_j\to {\mathbf{A}}_k^r$ whose components are the dehomogenized forms $f_1,\dots ,f_r$; its source and target are smooth over $k$ [F9], and its scheme-theoretic fibre over the origin is $Z(t)\cap X_j$ by 1.3. By [F36] the fibre tangent space at $x$ is $\ker d_{x}g$, and by [F37] the open immersion $Z(t)\cap X_j\subseteq Z(t)$ induces an isomorphism of tangent spaces, so ${\dim }_k\ker d_{x}g=d-r$; rank-nullity together with ${\dim }_k{T}_x{X}_j=d$ (from $\dim {\mathcal{O}}_{X_j,x}=d$ and [F15]) gives that $d_{x}g$ is surjective, of rank $r={\dim }_0{\mathbf{A}}_k^r$ [F20]. But then [F36] applies and shows that the fibre $Z(t)\cap X_j$ has a regular local ring at $x$ of dimension ${\dim }_x{X}_j-{\dim }_0{\mathbf{A}}_k^r=d-r$; since $Z(t)\cap X_j$ is an open subscheme of $Z(t)$, the local ring ${\mathcal{O}}_{Z(t),x}$ is regular of dimension $d-r$.

The good tuple lies in the full-rank locus. Since $r\le d$, 4.1 provides a tuple $t_0=(F_1,\dots ,F_r)$ with $Z=Z(t_0)$ nonempty, smooth over $k$ and of pure dimension $d-r$. By [F9] and [F10] each point $x\in Z$ has an affine neighbourhood on which $Z\to \mathrm{Spec}k$ is standard smooth at $x$ of some relative dimension $n$; by [F35] the module $\mathrm{\Omega }$ is free of rank $n$ there, and every component of that standard smooth affine neighbourhood has dimension $n$, while those components are nonempty open pieces of the components of $Z$, all of dimension $d-r$ [F19]; hence $n=d-r$ and ${\dim }_{\kappa (x)}({\mathrm{\Omega }}_{Z\mathrm{/}k}\otimes \kappa (x))=d-r$ for every $x\in Z$. Therefore no point of $Z$ has the defect of 3.2, and $t_0\in U_r^{\mathrm{\prime }}$.

Full-rank tuples with nonempty intersection are smooth. Let $t$ be a closed point of $U_r^{\mathrm{\prime }}$ and suppose $Z(t)\mathrm{\ne }\mathrm{\varnothing }$. By 4.2 every closed point of the finite-type $k$-scheme $Z(t)$ is a regular point. The regular locus of $Z(t)$ is open by [F13]; if its complement $S$ were nonempty, then $S$ with its reduced closed-subscheme structure would be a nonempty closed subscheme of ${\mathbf{P}}_k^N$ of finite type over $k$, so 1.4 applied to $S$ would produce a point closed in $S$, hence in $Z(t)$ because $S$ is closed in $Z(t)$, a contradiction. Hence $Z(t)$ is regular, so $Z(t)\to \mathrm{Spec}k$ is smooth by [F12] since $k$ is perfect [F11]; and $Z(t)$ is reduced because its local rings are regular, hence domains, by [F14].

Claim 2. Suppose $r>d$. By 4.1 with $r=d$ (and 2.1 when $d=0$) there is a tuple $F_1,\dots ,F_d$ with $Z_d=Z(F_1,\dots ,F_d)$ nonempty, smooth over $k$ and of pure dimension $0$; by [F38] the underlying set of $Z_d$ is finite, say $\{p_1,\dots ,p_q\}$. For each $l$ the forms of $S_{e_{d+1}}$ vanishing at $p_l$ form a proper linear subspace: some coordinate function $x_m$ is nonzero at the closed point $p_l$ [F3], and then $x_m^{e_{d+1}}$ does not vanish there [F4]. Since the field $k$ is infinite [F3], [F25] provides $F_{d+1}\in S_{e_{d+1}}$ vanishing at none of $p_1,\dots ,p_q$, and we choose arbitrary nonzero forms $F_{d+2},\dots ,F_r$ (for instance powers of coordinates), which exist because $S_e\mathrm{\ne }0$ for $e\ge 1$; then the underlying set of $Z(F_1,\dots ,F_r)$, being contained in $Z_d\cap V_{+}(F_{d+1})$, is empty, so $Z(F_1,\dots ,F_r)=\mathrm{\varnothing }$. Therefore the open set $\{t\in {\mathrm{\Pi }}_r:Z(t)=\mathrm{\varnothing }\}$ of 2.3 is nonempty, which is claim 2.

Full-rank tuples with nonempty intersection have pure dimension $d-r$. Let $t$ be a closed point of $U_r^{\mathrm{\prime }}$ with $Z(t)\mathrm{\ne }\mathrm{\varnothing }$; then $Z(t)$ is reduced by 5.2, so [F16] applies at every closed point $x$ of $Z(t)$ and, together with 4.2, gives that the maximum of $\dim W$ over the irreducible components $W$ of $Z(t)$ containing $x$ equals $\dim {\mathcal{O}}_{Z(t),x}=d-r$. Let $W$ be any irreducible component of $Z(t)$ (finitely many exist by [F17]): the open subset $W\setminus {\bigcup }_{W^{\mathrm{\prime }}\mathrm{\ne }W}{W}^{\mathrm{\prime }}$ of $W$ is nonempty, because otherwise the irreducible $W$ would be contained in the finite union of the closed sets $W^{\mathrm{\prime }}$ and hence in one of them by [F18], contradicting that components are maximal; by 1.4 applied in a chart meeting it, it contains a closed point $x$ of $Z(t)$, which then lies on no component other than $W$, so the maximum above is $\dim W$ and $\dim W=d-r$. Hence every irreducible component of $Z(t)$ has dimension $d-r$, i.e. $Z(t)$ is of pure dimension $d-r$.

Claim 1. Let $0\le r\le d$ and put $U_r=U_r^{\mathrm{\prime }}$, the open full-rank locus of 3.2. It is nonempty because it contains the tuple $t_0$ of 4.1 by 5.1. For every closed point of $U_r$ the intersection is nonempty by 2.3, smooth over $k$ by 5.2 and of pure dimension $d-r$ by 6.1; this is claim 1, and for $r=0$ it is also the statement of 2.1.

Boundary, choice, and iff dispositions. Empty: the statement has $X\mathrm{\ne }\mathrm{\varnothing }$; for $r\le d$ every member over $U_r$ is nonempty by 7.1, and for $r>d$ the members over the open set of 5.3 are empty, the empty scheme being allowed there. Zero: the case $r=0$ is the empty-tuple case of 2.1 with ${\mathrm{\Pi }}_0=\mathrm{Spec}k$, and $d=0$ is covered by 4.1 and 5.3; for $r=d$ the conclusion is pure dimension $0$, i.e. a finite nonempty set of closed points, consistent with [F38]. One: $r=1$, $1\le d$, is the first induction step of 4.1, and the parabolas/hypersurface computations of the companion page are instances; no step requires $r\ge 2$. Degenerate: $X$ is irreducible of pure dimension $d$ and smooth, so no singular-source case arises; the members $Z(t)$ are allowed to be reducible or non-reduced as subschemes of $X$, and no irreducibility, connectedness, or nonemptiness is asserted for tuples outside $U$. Endpoints: the degrees $e_i\ge 1$ and $N\ge 0$ are arbitrary; $d=0$, $r=0$, $r=d$ and $r>d$ are all covered, and for $r>d$ the statement covers every $r$, not merely $d+1$. Nonempty-choice: AC is declared as [F1] and is used exactly through the AC-assuming suppliers [F7] (dictionary), [F8] (closed-point density and Nullstellensatz), [F12]-[F13] (regular versus smooth, openness of the regular locus), [F16] (componentwise local dimension), [F26] (hypersurface dimension drop), [F29] (Bertini), [F31] (closedness of the projection), [F33] (Jacobian criterion), [F35]-[F36] (standard smooth fibres and the tangent criterion), and [F38] (zero-dimensional varieties are finite); the finite choices of charts, generating lists, components, coefficients and forms in steps 1.3, 2.3, 3.1, 3.2, 5.1, 5.3 and 6.1 are finite and add no choice principle, and the linear algebra and differential computations are choice-free. Both iff cases: the biconditional [F12] is used in the direction "smooth implies regular" in 1.1, 4.1 and 4.2 and in the direction "regular implies smooth" in 5.2; the criterion [F36] is used in the direction "surjective differential implies smooth at $x$" in 4.2 after the converse direction is only used through the kernel identification $T_x(\text{fibre})=\ker d_{x}f$, which [F36] supplies for every such morphism; the Jacobian criterion [F33] is used in the rational-point direction "regular implies the rank formula" in 4.2 and in the perfect-field direction only through the same equivalence; the irreducibility criterion [F24] is used in the direction "vanishing ideal $(0)$ prime implies ${\mathbf{P}}_k^n$ irreducible" in 1.2; and the Nullstellensatz facts [F8] are used in both directions in 1.4 to identify closed points with residue field $k$. No claim is made about the size or density of the open sets $U$, and the characteristic-$0$ hypothesis enters only through Bertini [F29] and the perfectness of $k$ [F11]. This completes the proof.