A dense hypersurface chart with a nonzero partial derivative

By [F3], choose a nonempty affine chart $U\subseteq X$. Its coordinate ring is finite type by [F4], so its fraction field is finitely generated over $k$; by [F5] this field is $K:=k(X)$. Equation [F6] gives ${\mathrm{trdeg}}_{k}K=d$.

By [F2] and [F7], choose a separating transcendence basis $t_1,\dots ,t_d$ of length $d$ for $K\mathrm{/}k$; its residual extension $K\mathrm{/}L$, for $L=k(t_1,\dots ,t_d)$, is finite separable. Here $A=k[T_1,\dots ,T_d]$ has fraction field $L$ under $T_i\mapsto t_i$. If $d=0$, then $K\mathrm{/}k$ is finite and every element has an irreducible minimal polynomial over the algebraically closed field $k$; [F2] and [F9] force each such polynomial to be linear, so $K=k$ and we may take $\gamma =0$. In general, [F8] gives $\gamma \in K$ with $K=L(\gamma )$; when $K=L$, again take $\gamma =0$.

Let $m(Z)\in L[Z]$ be the monic minimal polynomial of $\gamma$. It is irreducible by [F9] and separable by [F10], since $K\mathrm{/}L$ is separable. Hence $m^{\mathrm{\prime }}(Z)\mathrm{\ne }0$ by [F12]. For the trivial extension this is $m(Z)=Z$, so the same derivative conclusion holds. Also [F9] and [F11] identify $L[Z]\mathrm{/}(m)$ with the field $L(\gamma )=K$.

Clear the finitely many coefficient denominators of $m$ with a nonzero $q\in A$, obtaining $Q=qm\in A[Z]$. In the UFD $A$, factor the common irreducible divisors of the finitely many coefficients of $Q$ to write $Q=cP$, where $c\in A\setminus \{0\}$ is their common content and $P$ is primitive. The polynomial $P$ has positive $Z$-degree and is an associate of $m$ over $L$. Thus it is irreducible in $L[Z]$, and [F14] makes it irreducible in $A[Z]$. Since $q\mathrm{/}c\in L^{\times }$ is independent of $Z$, $P=(q\mathrm{/}c)m$ in $L[Z]$ and $P_Z=(q\mathrm{/}c)m^{\mathrm{\prime }}\mathrm{\ne }0$.

Put $H=V(P)\subseteq {\mathbf{A}}_k^{d+1}$. By [F13] the irreducible polynomial $P$ is prime in $A[Z]$; it is a nonunit because it has positive $Z$-degree. Therefore $(P)$ is a proper prime ideal. By [F15], $H$ is nonempty and irreducible and $I(H)=\sqrt{(P)}=(P)$. Thus [F4] gives $k[H]=A[Z]\mathrm{/}(P)$, a domain, and [F16] gives $k(H)=\mathrm{Frac}(A[Z]\mathrm{/}(P))$. Localizing this coordinate ring at the nonzero elements of $A$ yields $L[Z]\mathrm{/}(P)\cong L[Z]\mathrm{/}(m)\cong L(\gamma )=K$. Here $A$ embeds in $A[Z]\mathrm{/}(P)$ because $P$ has positive $Z$-degree, and the localized quotient is a field by [F11]; hence it is the fraction field of $A[Z]\mathrm{/}(P)$. Consequently $k(H)\cong k(X)$ over $k$.

Both $X$ and $H$ are integral classical varieties by their hypotheses and step 5.1. Their function fields are $k$-isomorphic, so [F17] and [F18] supply isomorphic nonempty open subvarieties. By [F6], $\dim H={\mathrm{trdeg}}_{k}k(H)=d$. Step 4.1 gives the required nonzero last-coordinate partial derivative, and the construction makes $H$ an irreducible hypersurface in ${\mathbf{A}}_k^{d+1}$. AC is propagated through the integral-atlas, chart-function-field, dimension, Nullstellensatz, affine-function-field and birational interfaces [F3, F5, F6, F15, F16, F17]; once these apply, the construction uses only the finite chart, basis, generator and denominator/content selections above.