TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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A minimal-parabolic flag projection is a projective-line bundle

Statement

Assume the Axiom of Choice. Let $G$ be the connected simply connected complex semisimple affine algebraic group with Borel $B = T ⋉ U$ and Weyl group $W$ of Complex semisimple algebraic group, Borel, and flag variety and Borel, opposite unipotent groups and root coordinates, and let $α$ be a simple root with minimal parabolic $P_{α} = B ⊔ B n_{α} B$ , root subgroup $U_{- α}$ and Weyl subgroup ${1, s_{α}}$ as in Minimal parabolic from one negative simple root. Let $π_{B} : G \to X_{B}$ and $π_{α} : G \to X_{α}$ be the orbit maps of Projective orbit constructions for G/B and G/P_alpha. Then:

(i) the induced map $f : X_{B} \to X_{α}$ , $g [v_{B}] \mapsto g [v_{α}]$ , is a surjective morphism of varieties, well defined because $B \subseteq P_{α}$ , and its fibre over $g [v_{α}]$ is canonically the coset space $P_{α} / B$ , which is $P^{1}$ in the two-chart description $z \mapsto u_{- α} (z) B$ , $t \mapsto u_{α} (t) n_{α} B$ with $t = z^{- 1}$ of Minimal parabolic from one negative simple root;

(ii) $f$ is a Zariski-locally trivial fibre bundle with fibre $P^{1}$ : over each single $G$ -translate $g V$ of the open torsor chart $V = σ_{α} (U_{α}^{-}) \subseteq X_{α}$ from Zariski sections of Borel and minimal-parabolic orbit maps, it becomes the projection $g V \times P^{1} \to g V$ . These translates cover $X_{α}$ and admit a finite subcover. In the rank-one case $G = P_{α}$ , the base is a point and $f$ is the unique map from $P^{1}$ to that point.

Facts & Assumptions

Given: the group $G$ , its Borel $B$ , the simple root $α$ , the minimal parabolic $P_{α}$ , the orbit maps $π_{B}$ , $π_{α}$ and the closed orbits $X_{B}$ , $X_{α}$ .

[F1]

$P_{α}$ is a closed connected subgroup containing $B$ and $U_{- α}$ with $P_{α} = B ⊔ B n_{α} B$ , $Lie P_{α} = b \oplus g_{- α}$ and $\dim P_{α} = \dim B + 1$ ; the coset space $P_{α} / B$ is described by the two affine charts $z \mapsto u_{- α} (z) B$ and $t \mapsto u_{α} (t) n_{α} B$ glued by $t = z^{- 1}$ . (Minimal parabolic from one negative simple root)

[F2]

The stabilizer of $[v_{B}]$ in $G$ is $B$ , the stabilizer of $[v_{α}]$ is $P_{α}$ , and the fibres of $π_{B}$ and $π_{α}$ are exactly the right cosets of $B$ respectively $P_{α}$ ; both orbit maps are surjective onto the closed orbits $X_{B} = π_{B} (G)$ , $X_{α} = π_{α} (G)$ , which are smooth projective of dimensions $∣ Φ^{+} ∣$ and $∣ Φ^{+} ∣ - 1$ . (Projective orbit constructions for G/B and G/P_alpha)

[F3]

The orbit map $π_{α} : G \to X_{α}$ is a Zariski-locally trivial right $P_{α}$ -torsor. Its open chart $V = σ_{α} (U_{α}^{-})$ satisfies $π_{α}^{- 1} (V) ≅ V \times P_{α}$ , and the single $G$ -translates $g V$ cover $X_{α}$ , with the product torsor transported to each translate. Likewise $π_{B}$ is a right $B$ -torsor and represents the fppf sheaf quotient $G / B$ . (Zariski sections of Borel and minimal-parabolic orbit maps)

[F4]

The Axiom of Choice is The Axiom of Choice.

Proof

1.1F1F2F3construct

The inclusion $B \subseteq P_{α}$ makes $g [v_{B}] \mapsto g [v_{α}]$ well defined on closed points, and $π_{α}$ is surjective by [F2]. It is a morphism: by [F3] a Zariski cover of $X_{B}$ admits sections of the $B$ -torsor $π_{B}$ , so on each chart the proposed map is the composite of a section into $G$ with the morphism $π_{α}$ ; the expressions agree on overlaps because two sections differ by right multiplication by a $B$ -valued function and $B \subseteq P_{α}$ . The local morphisms glue to $f$ , and $f \circ π_{B} = π_{α}$ as morphisms.

1.2F1F2F3

Fix a point $y = g [v_{α}] \in X_{α}$ . Since $π_{α}^{- 1} (y) = g P_{α}$ by [F2], the fibre of $f$ is the image of $g P_{α}$ under $π_{B}$ . The quotient description in [F3], after choosing the displayed $g$ , identifies this fibre as a scheme with $P_{α} / B$ . By [F1] the latter fppf quotient is $P^{1}$ with charts $z \mapsto u_{- α} (z) B$ , $t \mapsto u_{α} (t) n_{α} B$ and transition $t = z^{- 1}$ .

2.1F1F2F3step 1.1step 1.2construct

Local triviality follows by base change of the $P_{α}$ -torsor. Let $V = σ_{α} (U_{α}^{-})$ . The product chart of [F3] identifies $π_{α}^{- 1} (V)$ with $V \times P_{α}$ , equivariantly for right $P_{α}$ . Quotienting this identity by the right subgroup $B$ gives $f^{- 1} (V) ≅ (V \times P_{α}) / B ≅ V \times (P_{α} / B) ≅ V \times P^{1}$ as fppf sheaves and hence as schemes by [F1] and [F3]. Under this identification $f$ is the projection to $V$ . For every $g \in G$ , left translation carries the entire diagram to the single open translate $g V$ and gives the same product description. These translates cover $X_{α}$ by [F3], and projectivity makes a finite subcover available.

2.2F1F2step 1.2

The rank-one case. If $G = P_{α}$ then $B \subseteq P_{α} = G$ and $X_{α} = G / P_{α}$ is a single point while $X_{B} = G / B = P_{α} / B$ is $P^{1}$ by [F1] and [F2]; the map $f$ is then the unique morphism $P^{1} \to {p t}$ , which is the trivial $P^{1}$ -bundle over its one-point base, and both assertions hold without any local section.

3.1F1F2F3F4step 1.1step 1.2step 2.1step 2.2discharge-construct∎

Steps 1.1–1.2 prove (i), and step 2.1 proves (ii); step 2.2 checks the rank-one endpoint. The Axiom of Choice is inherited through [F1]–[F3] and declared in [F4].

Depends on

Used by

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Groups (standard reference, not scraped)
Michel Brion, Lectures on the Geometry of Flag Varieties (standard reference, not scraped)