Bertini curve functional with a prescribed avoidance locus

PDF p. 15: Theorem6.7 second Bertini application

Assume AC and the preceding point-section construction over an algebraically closed field $k$ of any characteristic. First suppose the normal affine threefold $X$ is integral. Let $U=X\setminus S$ and $Q{\mathrm{\mid }}_U$ be its rank-two kernel bundle. For every fixed closed subset $Z\subset U$ of codimension at least two, there exists a functional $t:Q{\mathrm{\mid }}_U\to {\mathcal{O}}_U$ whose scheme-theoretic zero locus $Y$ is nonempty, geometrically integral, smooth over $k$, and of pure codimension two in $U$, and satisfies $Y\cap Z=\mathrm{\varnothing }$. In particular it is smooth outside $U_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$, and can be chosen entirely inside $U_{\mathrm{s}\mathrm{m}\mathrm{/}k}$ by adjoining $U_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ to the fixed avoidance set. The parameter may depend on $Z$.

For a normal affine threefold understood as a finite disjoint union of integral threefolds, apply this assertion on each clopen component: there is a single functional with zero scheme a finite disjoint union of nonempty smooth geometrically integral codimension-two curves, one per component, avoiding the prescribed $Z$. Global irreducibility is asserted in the integral case, not for every arbitrary fixed pair $(U,Q)$ on a disconnected scheme; the counterexample and exact scope correction are proved in the component lemma. The global theorem6.7 can be reduced to these components and does not require a new connectedness hypothesis. If “threefold” is interpreted as allowing additional lower-dimensional components, the curve assertion concerns its three-dimensional clopen components; the theorem's lower-dimensional cases must be handled in their own dimension, rather than falsely called curves here.

The article's existence sentence and condition(4) are read in their ordinary prescribed-locus sense. Taking $Z=Y$ after choosing $t$ reverses the intended quantifiers and is not an original-paper counterexample. KM82 section3 full text was not read; the substantial external theorem actually used is Swan1974 Theorem1.4 and its explicit controlling-family proof, read in the original scans. The support consumer may use the integral case on each clopen component, or carry the finite union; it must not silently assume a globally irreducible curve on every disconnected fixed pair.