Curve-and-point support reduction

By [F1], choose a representative $P_0$ of $\alpha$ with the stated rank, determinant and Chern properties. By [F2], each lower-dimensional clopen component has zero third filtration image, so $\alpha$ restricts to zero there. Replace $P_0$ on all such components by ${\mathcal{O}}^3$, leaving it unchanged on the three-dimensional components. Gluing over the finite disjoint clopen partition gives a rank-three bundle $P$. The componentwise $K_0$ decomposition in [F2] shows that $[P]-[{\mathcal{O}}_X^3]=\alpha$. Its determinant and the two Chern vanishings hold componentwise, hence globally.

Apply the point-section supplier [F3] to the restriction of $P$ on the union of three-dimensional components. On each lower-dimensional component use the first-coordinate projection ${\mathcal{O}}^3\to \mathcal{O}$. These maps glue to $s$. Its zero scheme $S$ is finite, lies in the smooth locus, and has no points on lower-dimensional components. On $U=X\setminus S$, [F3] gives the exact sequence $0\to Q{\mathrm{\mid }}_U\to P{\mathrm{\mid }}_U\stackrel{s}{\to }{\mathcal{O}}_U\to 0$ with rank-two kernel; on lower-dimensional components this kernel is the displayed trivial rank-two summand.

On each three-dimensional component, prescribe the closed nonsmooth locus in $U$ as the avoidance set. It has codimension at least two by [F3]. The curve supplier [F3] gives a functional on the actual restricted kernel whose zero scheme is a smooth geometrically integral codimension-two curve avoiding that set. On each lower-dimensional component use a coordinate projection from the trivial rank-two kernel; it has empty zero scheme. Glue these finitely many functionals to $t$ on $U$. Thus $Y$ is precisely the stated finite disjoint union of curves and lies in the smooth locus. The construction uses the avoidance set fixed before choosing the functional.

The zero scheme $Y$ is closed in the open subset $U$. Its reduced closure $C$ in $X$ therefore satisfies $C\cap U=Y$ as reduced schemes: $Y$ is smooth and hence reduced. Thus $C\setminus Y\subseteq X\setminus U=S$. Since $Y$ is smooth and every point of $S$ is in the smooth locus of $X$, both $C$ and $T=C\cup S$ have the asserted location properties; the possible nonsmooth points of $C$ are contained in $C\setminus Y$. This also applies to the empty boundary.

On $V=X\setminus (C\cup S)$ both functionals are pointwise nonzero. A functional on a finite locally free module is surjective wherever one of its coefficients is a unit, so the sequence for $s$ restricts to the first displayed sequence and $t{\mathrm{\mid }}_V$ is surjective. By [F4], $\det Q{\mathrm{\mid }}_V\otimes \det {\mathcal{O}}_V\cong \det P{\mathrm{\mid }}_V\cong {\mathcal{O}}_V$. The rank-two kernel result in [F4] identifies the kernel of $t{\mathrm{\mid }}_V$ with ${\mathcal{O}}_V$, giving the second sequence.

Additivity of bundle classes in perfect $K_0$ gives $[Q{\mathrm{\mid }}_V]=2[{\mathcal{O}}_V]$ and then $[P{\mathrm{\mid }}_V]=3[{\mathcal{O}}_V]$, so $\alpha {\mathrm{\mid }}_V=0$. The finite-type affine scheme $X$ is Noetherian and qcqs; its open subset $V$ is quasi-compact. Applying the actual localization fiber sequence in [F5], exactness on zeroth homotopy identifies the kernel of $K_0(X)\to K_0(V)$ with the image of $K_{0,T}(X)\to K_0(X)$. Therefore $\alpha$ has the claimed supported lift. The lift is an existence statement and is not asserted to be canonical.