Rank-two Chern map and its fixed-invariant fibers

Assume the Axiom of Choice and the supplied universe conventions. Let $X$ be a smooth affine scheme of finite type and pure dimension three over a field. Write $V={\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_2(X)$ for the set of isomorphism classes of rank-two finite locally free sheaves. The determinant of $E$ is the invertible sheaf $\det E={\bigwedge }^{2}E$. The group $\mathrm{Pic}(X)$ is the group of isomorphism classes of invertible sheaves, under tensor product, with identity $[{\mathcal{O}}_X]$. The group ${\mathrm{C}\mathrm{H}}^i(X)$ is the free abelian group on integral codimension-$i$ closed subschemes modulo rational equivalence: the subgroup generated by divisors of rational functions on integral codimension-$(i-1)$ closed subschemes. Use the supplied Chow-theoretic Chern classes, and use the determinant convention for the first invariant in $\mathrm{Pic}(X)$. Define $C_X:V⟶\mathrm{Pic}(X)\times {\mathrm{C}\mathrm{H}}^2(X),[E]⟼([\det E],c_2(E))\mathrm{.}$ For a line-bundle class $L$ put $V_L=\{e\in V:[\det e]=L\}$. For $a\in {\mathrm{C}\mathrm{H}}^2(X)$ put $V_{L,a}=\{e\in V_L:c_2(e)=a\}$. These are sets of bundle isomorphism classes, without choices of a determinant trivialization. The target basepoint is $([{\mathcal{O}}_X],0)$ and the source basepoint is $[{\mathcal{O}}_X^{\oplus 2}]$.

Isomorphism classes of vector bundles

Chern classes of a vector bundle on a smooth scheme

The Axiom of Choice

1.1 Applying the second exterior-power functor to the local rank-two trivializations of $E$ gives rank-one trivializations with transition functions the determinants of the original matrices. The determinant identity for products verifies the cocycle condition. An isomorphism of bundles induces an isomorphism of their second exterior powers. Thus $[\det E]$ depends only on $[E]$. Tensor products and duals of invertible sheaves respectively give the stated group law and inverses on their isomorphism classes. [F1, construct]

2.1 The supplied Chern construction gives $c_2(E)\in {\mathrm{C}\mathrm{H}}^2(X)$ invariant under bundle isomorphism. Combining this with step 1.1 defines $C_X$ on the set $V$; separation defines the subsets $V_L$ and $V_{L,a}$. The determinant of the trivial rank-two bundle is trivial and its second Chern class is zero, so the chosen basepoints are respected. [F1, F2, step 1.1] ∎

The notation makes explicit the first coordinate of Asok–Fasel Theorem 1. No assertion that arbitrary pairs are realized, or uniquely realized, is part of this definition. The identification of the determinant coordinate with the usual $c_1$ coordinate in ${\mathrm{C}\mathrm{H}}^1$ is included in the external comparison input below, rather than derived from a statement about abstractly isomorphic groups.