Bertini functional with finite smooth zero locus

PDF p. 15: Theorem6.7: rank-three Bertini functional

Assume AC. Let $X$ be a normal affine threefold of finite type over an algebraically closed field $k$ of arbitrary characteristic, and let $P$ be a rank-three vector bundle on $X$. There is a functional $s:P\to {\mathcal{O}}_X$ whose scheme-theoretic zero locus $S=V(\mathrm{im}s)$ is a finite smooth $k$-scheme contained in $X_{\mathrm{s}\mathrm{m}\mathrm{/}k}$. Thus $S$ is a finite set of distinct reduced $k$-rational closed points, possibly empty. No Chern-class or determinant hypothesis on $P$ is needed for this assertion; it therefore applies to the particular $P$ in Theorem6.7. Threefold may be read as pure dimension three or, more generally, a normal scheme of dimension three; the construction below treats lower-dimensional clopen components as an additional avoidance locus.

Let $Q=\ker s$. It is coherent, and there is a surjection ${\mathcal{O}}_X^m\twoheadrightarrow Q$ for some finite $m$. On $U=X\setminus S$, the restriction $Q{\mathrm{\mid }}_U$ is a rank-two vector bundle, its displayed generators restrict to a surjection ${\mathcal{O}}_U^m\twoheadrightarrow Q{\mathrm{\mid }}_U$, and $0⟶Q{\mathrm{\mid }}_U⟶P{\mathrm{\mid }}_U\stackrel{s{\mathrm{\mid }}_U}{\to }{\mathcal{O}}_U⟶0$ is exact. Hence $Q{\mathrm{\mid }}_U$ is generated by global sections on $U$, even though $U$ need not be affine.

The article cites Swan's Bertini form through KM82 section3. The authored proof here instead gives the characteristic-free finite-first-jet incidence argument, following the bad-locus method actually read in Stacks Lemma33.47.3 and extending it from a line-bundle hyperplane family to the explicit rank-three determinant condition. It neither attributes this proof to unread KM82 nor assumes the characteristic-zero conventions of another Swan formulation. There is no claim about the second, rank-two irreducible-curve Bertini application later in the article's paragraph; that is outside this assigned proposition.