Bertini smoothness away from the base locus

Setup, indexing, and the parameter space. Put $r={\dim }_{k}W-1\ge 0$, so that $\mathbf{P}(W)={\mathbf{P}}_k^r$ by [F2] and [F3], and let $I$, $\pi \text{\hspace{0.17em}⁣}:I\to X^{\circ }$, $p\text{\hspace{0.17em}⁣}:I\to \mathbf{P}(W)$ be the incidence of [F3]; by [F3] clause (3) the morphism $I\to \mathrm{Spec}k$ is smooth, so $I$ is a finite-type $k$-scheme, and by the construction recorded in [F3] the scheme $I$ is a locally closed subscheme of $X^{\circ }{\times }_k{\mathbf{P}}_k^r$.

The k-rational parameter space is an irreducible classical projective variety. By [F25] the space ${\mathbf{P}}_k^r$ is nonempty, and ${\mathbf{P}}_k^r=V_{+}((0))$ is a projective algebraic set. Its homogeneous vanishing ideal is $I_{+}({\mathbf{P}}_k^r)=(0)$: if $0\mathrm{\ne }F\in k[x_0,\dots ,x_r]$ is homogeneous of positive degree and vanished at every point of ${\mathbf{P}}_k^r$, then the polynomial $F\in k[x_0,\dots ,x_r]$ would vanish at every point of $k^{r+1}$ (a nonzero point $a$ gives $[a]$, and $F(0)=0$ in positive degree), so $F=0$ by [F29] applied with $R=S=k$ (the algebraically closed field $k$ is infinite) and $m=r+1\ge 1$, a contradiction. Since $(0)$ is prime by [F28] and $k[x_0,\dots ,x_r]$ is the homogeneous coordinate ring of [F26], [F27] shows that ${\mathbf{P}}_k^r$ is irreducible; by [F30] it is a classical projective variety. Moreover ${\mathbf{P}}_k^r$ is an integral finite-type $k$-scheme: its standard affine charts are spectra of polynomial rings over $k$ by [F38], which are domains by [F28], so the nilradical ideal sheaf of [F9] vanishes on a chart cover and ${\mathbf{P}}_k^r$ is reduced, and it is finite type over $k$ because the charts of [F38] give a finite affine cover by finitely generated $k$-algebras [F12].

The incidence is regular, reduced and Noetherian. By [F3] clause (3) and [F8], the finite-type $k$-scheme $I$ is smooth over the perfect field $k$, so [F7] makes $I$ regular: every local ring ${\mathcal{O}}_{I,x}$ is a regular local ring. Each such ring is a domain by [F6], hence reduced; therefore the nilradical ideal sheaf ${\mathcal{N}}_I$ of [F9] has zero stalks, $I=I_{\mathrm{r}\mathrm{e}\mathrm{d}}$, and $I$ is a reduced scheme. By [F12] the finite-type morphism $I\to \mathrm{Spec}k$ is quasi-compact and locally of finite type, so $I$ has a finite affine open cover by spectra $\mathrm{Spec}{A}_j$ of finitely generated $k$-algebras $A_j$; each $A_j$ is Noetherian by [F10] since the field $k$ is Noetherian, so $I$ is a Noetherian scheme by [F11]. The underlying space $\mathrm{\mid }I\mathrm{\mid }$ is a Noetherian topological space: each $\mathrm{Spec}{A}_j$ is Noetherian by [F39], and a descending chain of closed subsets of $I$ restricts to descending chains in the finitely many charts, each of which stabilizes, whence the chain itself stabilizes.

Separatedness of the components. Since $X$ admits a locally closed immersion into a projective space [F22], $X\to \mathrm{Spec}k$ is separated: the immersion is separated by [F19], the projective space is separated over $\mathrm{Spec}k$ by [F18], and separated morphisms compose by [F20]. The open subscheme $X^{\circ }\subseteq X$ is separated over $\mathrm{Spec}k$ by [F19] and [F20], ${\mathbf{P}}_k^r\to \mathrm{Spec}k$ is separated by [F18], so $X^{\circ }{\times }_k{\mathbf{P}}_k^r\to \mathrm{Spec}k$ is separated by [F21] and [F20]; the locally closed subscheme $I$ of [F3] is therefore separated over $\mathrm{Spec}k$ by [F19] and [F20].

The hyperplane system and its base locus. Suppose now that $X\mathrm{\ne }\mathrm{\varnothing }$, fix the locally closed immersion $X\hookrightarrow {\mathbf{P}}_k^N$, let $L={\mathcal{O}}_X(1)$ and let $W_h\subseteq \mathrm{\Gamma }(X,{\mathcal{O}}_X(1))$ be the hyperplane system of [F2]. Then $W_h\mathrm{\ne }0$: if the restriction of every degree-one form vanished on $X$, then $x_0,\dots ,x_N$ would all vanish on $X$, whence $X\subseteq V_{+}(x_0,\dots ,x_N)=\mathrm{\varnothing }$ by [F25], contradicting $X\mathrm{\ne }\mathrm{\varnothing }$. Also $\mathrm{Bs}(W_h)=\mathrm{\varnothing }$: for every point $x\in X$ some standard chart $D_{+}(x_j)$ of ${\mathbf{P}}_k^N$ contains $x$ by [F38], and on that chart the restricted linear form $x_j{\mathrm{\mid }}_X$ is a unit at $x$, so its zero subscheme does not contain $x$ and $x\mathrm{\notin }\mathrm{Bs}(W_h)$ by [F2].

The finite component decomposition. By [F13] and step 1.3 the scheme $I$ has only finitely many irreducible components $Z_1,\dots ,Z_m$; each $Z_i$ is closed by [F14], and every point of $I$ lies on at least one $Z_i$ by [F14]. By [F5] and step 1.3 every point of $I$ lies on exactly one irreducible component, so the $Z_i$ are pairwise disjoint; since they are finitely many closed pairwise disjoint subsets, the complement of $Z_i$ is the union of the remaining closed $Z_j$, hence $Z_i$ is also open in $I$. Give $Z_i$ the open subscheme structure. Then each $Z_i$ is irreducible and, as an open subscheme of the reduced scheme $I$, reduced, hence integral by [F15]; it is finite type over $k$ as an open subscheme of the finite-type $k$-scheme $I$, smooth over $k$ because smoothness is local on the source [F32], and separated over $k$ because it is an open subscheme of the separated scheme $I$ of step 1.4, using [F19] and [F20].

The components and the parameter space as classical varieties. Each $Z_i$ of step 2.1 is an integral finite-type $k$-scheme, and by [F17] and [F16] the separatedness of $Z_i\to \mathrm{Spec}k$ from step 2.1 is exactly the affine-overlap separation condition; hence by [F24] and [F23] $Z_i$ corresponds to an irreducible classical variety over $k$, with classical points the closed points and with scheme $k$-morphisms corresponding to regular maps. Similarly $\mathbf{P}(W)={\mathbf{P}}_k^r$ is an integral finite-type $k$-scheme by step 1.2 and separated over $k$ by [F18], so by [F24] and [F23] it is an irreducible classical variety whose classical points are its closed points, and the restriction $p_i=p{\mathrm{\mid }}_{Z_i}\text{\hspace{0.17em}⁣}:Z_i\to \mathbf{P}(W)$ of [F3] is a $k$-morphism of schemes, hence a morphism of classical varieties under [F24].

Target generic smoothness on the dominant components. Let $i\in \{1,\dots ,m\}$ be such that $p_i$ is dominant in the sense of [F37]. By step 3.1 the source $Z_i$ and the target $\mathbf{P}(W)$ are irreducible classical varieties, $p_i$ is a morphism of classical varieties, and $Z_i$ is smooth over $k$; so [F4] clause (2) applies and produces a nonempty open subvariety $V_i\subseteq \mathbf{P}(W)$ such that for every closed point $y\in V_i$, equivalently every classical point of $V_i$ by [F24], the scheme-theoretic fibre $p_i^{-1}(y)=Z_i{\times }_{\mathbf{P}(W)}\mathrm{Spec}k(y)$ of [F34] is nonempty, smooth over $k$, and of pure dimension $\dim Z_i-r$.

The non-dominant components. For the component morphism $p_i$ of step 3.1, if it is not dominant, then by [F37] the image $p_i(Z_i)$ is not dense in $\mathbf{P}(W)$, so its closure is a proper closed subset and $W_i:=\mathbf{P}(W)\setminus \overline{p_i(Z_i)}$ is a nonempty open subset of $\mathbf{P}(W)$; by definition of the image, every point $y\in W_i$ has empty fibre $p_i^{-1}(y)=\mathrm{\varnothing }$.

The common parameter open set. There are finitely many components, so the family of nonempty open sets consisting of the $V_i$ of step 4.1 for the dominant components and the $W_i$ of step 4.2 for the non-dominant components is finite; let $U$ be their intersection, an open subset of $\mathbf{P}(W)$. By steps 1.2 and [F31], $\mathbf{P}(W)$ is irreducible, so any two of these nonempty open sets meet and, by induction on the finite list, $U\mathrm{\ne }\mathrm{\varnothing }$; if $I=\mathrm{\varnothing }$, so that there are no components, take $U=\mathbf{P}(W)$. In either case $U$ is a nonempty open subset of $\mathbf{P}(W)$. Distinct $Z_i$ are disjoint by step 2.1, so for every point $[s]\in U$ exactly one alternative of steps 4.1 and 4.2 applies to each component.

Smoothness of the incidence fibres over $U$. Fix a closed point $[s]\in U$ of $\mathbf{P}(W)$ and write $F=p^{-1}([s])=I{\times }_{\mathbf{P}(W)}\mathrm{Spec}k([s])$ for the scheme-theoretic fibre of [F34]; here $k([s])=k$ by [F24], since classical points correspond to closed points and all classical points have residue field $k$. Because the pairwise disjoint open subschemes $Z_i$ cover $I$ by step 2.1, the open subschemes $F{\times }_I{Z}_i$ cover $F$, and by [F33] each $F{\times }_I{Z}_i$ is canonically identified with the fibre $p_i^{-1}([s])=Z_i{\times }_{\mathbf{P}(W)}\mathrm{Spec}k$ of $p_i$. For a dominant $i$, with $[s]\in V_i$, this fibre is nonempty and smooth over $k$ by step 4.1; for a non-dominant $i$, with $[s]\in W_i$, it is empty by step 4.2, and the empty scheme is smooth over $k$. Smoothness is local on the source by [F32], so $F$ is smooth over $k$.

Identification with the general member. By [F3] clause (2), the fibre $F=p^{-1}([s])$ of step 6.1 is isomorphic over $X^{\circ }$ to the zero subscheme $Z(s)\cap X^{\circ }$ of the section $s$ restricted to $X^{\circ }$ ([F36] and [F2]); hence $Z(s)\cap X^{\circ }$ is smooth over $k$. This holds for every closed point $[s]\in U$ and every representative $0\mathrm{\ne }s\in W$, since $Z(s)$ depends only on $[s]$ by [F2]. That is the first assertion of the statement.

Non-vacuity of the parameter set in the classical reading. The set $U$ of step 5.1 is a nonempty open subset of the projective space $\mathbf{P}(W)$ over the algebraically closed field $k$, and the standard charts $D_{+}(x_j)$ of [F38] cover it, so $U\cap D_{+}(x_j)$ is a nonempty open subset of an affine space ${\mathbf{A}}_k^r$ for some $j$; by [F38] it contains a closed point of that affine spectrum, which by [F38] is a classical point of ${\mathbf{P}}_k^r$, hence by [F24] a closed point of the scheme $\mathbf{P}(W)$. Thus $U$ contains closed points and the general-member statement of step 7.1 is not vacuous; in the classical dictionary of [F24] these are exactly the parameters $[s]\in U$.

Hyperplane sections of the fixed embedding. Here $X^{\circ }=X\setminus \mathrm{Bs}(W_h)=X$, so the first assertion, which is established by the argument of steps 1.1-7.1 applied with $W=W_h$ and $X^{\circ }=X$, gives a nonempty open $U\subseteq \mathbf{P}(W_h)$ such that for every closed point $[s]\in U$ the zero scheme $Z(s)\subseteq X$ is smooth over $k$. Let $F$ be any degree-one form with $F{\mathrm{\mid }}_X=s\mathrm{\ne }0$; on a standard affine chart $V=\mathrm{Spec}A\subseteq X$ trivializing ${\mathcal{O}}_X(1)$, the zero scheme $Z(s)$ is cut out by the local equation $f$ of $s$ ([F36]) and the scheme-theoretic intersection $X{\times }_{{\mathbf{P}}_k^N}{V}_{+}(F)$ is cut out by the same equation, because $V_{+}(F)$ is defined by the dehomogenized form $F$ and $f$ is its restriction to the chart; so the two closed subschemes of $X$ agree by [F35] and [F36]. Hence the scheme-theoretic hyperplane sections of $X$ for parameters in $U$ are smooth over $k$, which is the final assertion.

Boundary, choice, and scope dispositions. Empty: if $X=\mathrm{\varnothing }$ then $\mathrm{\Gamma }(X,L)=0$ and no nonzero $W$ exists, so the theorem is vacuous; if $X\mathrm{\ne }\mathrm{\varnothing }$ but $X^{\circ }=\mathrm{\varnothing }$, then by [F3] clause (2) every fibre $p^{-1}([s])=Z(s)\cap X^{\circ }$ is empty and smooth, and one may take $U=\mathbf{P}(W)$ in step 5.1, so the statement holds; [F3] clause (4) and the same fibre identification give the parallel empty-member conclusion when $r=0$. Zero: the parameter space is ${\mathbf{P}}_k^0$ when ${\dim }_{k}W=1$, and its single member is empty on $X^{\circ }$ by the previous sentence; conversely, the incidence $I$ itself can be empty exactly when $X^{\circ }=\mathrm{\varnothing }$ or $r=0$, and in both cases the argument of step 5.1 uses the empty family of components. One: the case ${\dim }_{k}W=2$, $r=1$, is included; nothing in the proof requires $r\ge 2$. Degenerate: $X$ is assumed neither irreducible nor connected nor of pure dimension, and the argument decomposes $I$ rather than $X$; the members $Z(s)\cap X^{\circ }$ may be reducible, empty, or non-reduced as ambient data, and no smoothness of $X^{\circ }$ outside $X$ is used. Endpoints: the proof covers $N=0$ in the final clause (where $X={\mathbf{P}}_k^0$ and $W_h$ is one-dimensional, $\mathrm{Bs}(W_h)=\mathrm{\varnothing }$) and imposes no upper bound on $N$ or on $r$; the claimed open set may be all of the base-locus-free parameter space, and no density or dimension of the good locus beyond nonemptiness openness is asserted. Nonempty-choice: AC is declared in [F1] and is used exactly through the AC-assuming suppliers [F3] (incidence), [F4] (generic smoothness), [F5] (one-component lemma), [F7] (regularity versus smoothness), [F10] (Noetherianity routes), [F13]-[F14] (finitely many components), [F24] (the classical-scheme dictionary), [F38]-[F39] (closed points and Noetherian spectra), and [F18]-[F21] (separatedness); the finite choices of charts, bases and component indices and the fibre computations of steps 2.1, 6.1, 7.1, 1.5 and 8.2 are finite and add no choice principle. Both iff cases: the only biconditional invoked as a supplier is [F7] (regular if and only if smooth over the perfect field $k$), used in step 1.3 in the direction "smooth over $k$ implies regular"; the criterion [F17] is used in the direction "separated implies the affine-overlap condition" in step 3.1; and [F27] is used in the direction "the vanishing ideal is prime implies irreducibility" in step 1.2. No irreducibility, connectedness, dimension, or nonemptiness of the members is asserted, in accordance with the statement. This completes the proof.