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Surface uniqueness from the cylinder and cancellation

Statement

Assume AC. Let $k$ be an algebraically closed field, let $X$ be a smooth affine finite-type $k$ -scheme of pure dimension two, and put $Y = X \times_{k} A_{k}^{1}$ . Suppose rank-three bundles on $Y$ with equal ordinary Chow Chern triples are isomorphic. If rank-two bundles on $X$ cancel a common trivial line, then rank-two bundles on $X$ with the same first two Chern classes are isomorphic.

Facts & Assumptions

Given: The data and hypotheses in the statement.

Goal: Establish the stated conclusion.

[F1]

The cylinder is a smooth affine threefold and its zero section retracts projection. Cylinder and zero section of a smooth affine surface

[F2]

Stabilization and pullback preserve the relevant Chern data, with third class zero. Both schemes are smooth equidimensional and finite type over $k$ , and projection and the zero section are $k$ -morphisms, as required by this supplier. Chern data survive addition of trivial bundles

[F3]

The maps use the ordinary Chow classes, with equality interpreted coefficientwise. The ordinary Chow Chern map on isomorphism classes

Proof

1.1F1F2F3given

Let $E, F$ have equal first two classes. By [F2], $p^{*} (E \oplus O_{X})$ and $p^{*} (F \oplus O_{X})$ have identical Chern triples. The assumed rank-three uniqueness on $Y$ gives an isomorphism between them.

2.1F1givenstep 1.1∎

Pull back along $s$ and use $s^{*} p^{*} = id$ to obtain $E \oplus O_{X} ≅ F \oplus O_{X}$ . The assumed cancellation of a common trivial line yields $E ≅ F$ .

Depends on

Used by · 1

Classical smooth critical-rank classifications

Dependency tree · 3 levels

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