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arxiv: 2604.00751 · v2 · submitted 2026-04-01 · 🧮 math.AG · math.RT

Truncated Grassmannians, blow-ups along Schubert varieties and collineations

Evgeny Feigin This is my paper

Pith reviewed 2026-05-13 22:19 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords truncated Grassmanniansblow-upsSchubert varietiesflag varietiescollineationsGrassmanniansorbit closureswedge powers
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The pith

Truncated Grassmannians describe the blow-ups of flag varieties along Schubert subvarieties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines truncated Grassmannians as the closures of orbits of abelian unipotent groups acting on degree truncations of projectivized wedge powers. It shows that these same truncations appear in the geometry of blow-ups of general flag varieties along Schubert subvarieties. The detailed case for Grassmannians places the blow-ups inside a larger family of varieties that project onto the Grassmannian, with fibers given by spaces of collineations.

Core claim

Truncated Grassmannians arise as the natural geometric objects that capture the local structure of blow-ups of flag varieties along Schubert subvarieties; for Grassmannians these blow-ups belong to a family of varieties projecting onto the base Grassmannian whose fibers are parametrized by spaces of collineations.

What carries the argument

Truncated Grassmannians, defined as closures of orbits of abelian unipotent groups on degree truncations of projectivized wedge powers, that encode the exceptional geometry of the blow-up along a Schubert subvariety.

If this is right

  • The blow-up of any flag variety along a Schubert subvariety admits a description in terms of truncated Grassmannians.
  • For Grassmannians the blow-up sits inside an explicit family of varieties that project onto the original Grassmannian.
  • The fibers of these projections are spaces of collineations between the relevant linear spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same truncated-orbit construction may supply local models for blow-ups along Schubert varieties in other homogeneous spaces beyond Grassmannians and full flags.
  • Collineation spaces appearing as fibers suggest a possible link between the blow-up geometry and the automorphism group of the underlying vector space.
  • The projection family offers a concrete way to deform the blow-up while keeping the base Grassmannian fixed, which could be used to study deformation theory or moduli of such resolutions.

Load-bearing premise

The orbit closures that define the truncated Grassmannians on truncated wedge powers accurately reproduce the local geometry of an arbitrary blow-up along a Schubert subvariety inside the flag variety.

What would settle it

Compute the blow-up of a low-dimensional Grassmannian along a concrete Schubert variety, extract the fiber over the base point, and check whether its structure matches the predicted truncated Grassmannian and collineation space.

read the original abstract

Truncated Grassmannians are defined as closures of orbits of abelian unipotent groups acting on the degree truncations of projectivized wedge powers. We show that such truncations in a more general setup show up in the description of the blow-ups of general flag varieties along Schubert subvarieties. We work out the case of Grassmannians in detail. In particular, we show that our blow-ups are members of a larger family of varieties projecting onto Grassmannians, and describe the fibers of these projections via the spaces of collineations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper defines truncated Grassmannians as the closures of orbits of abelian unipotent groups acting on degree-truncated projectivized wedge powers. It shows that these objects arise naturally in the blow-ups of general flag varieties along Schubert subvarieties, with a detailed treatment of the Grassmannian case. The blow-ups are embedded in a larger family of varieties that project onto Grassmannians, and the fibers of these projections are described in terms of spaces of collineations.

Significance. If the identifications are correct, the work supplies an explicit geometric model for blow-ups along Schubert varieties via truncated orbit closures and collineation spaces. This could furnish new tools for studying normal cones, resolutions, and degenerations in flag varieties. The detailed Grassmannian analysis and the uniform description across flag varieties constitute the main strengths.

major comments (1)
  1. The central claim equates the blow-up along an arbitrary Schubert subvariety with a member of the family whose exceptional locus is an orbit closure on a truncated wedge power. It is not clear from the setup whether a fixed truncation degree uniformly reproduces the full ideal sheaf of the Schubert variety, including higher syzygies beyond the first-order normal directions. A concrete verification for at least one non-trivial Schubert class in the Grassmannian case (e.g., via explicit ideal generators) is needed to confirm the identification holds without missing relations.
minor comments (2)
  1. Clarify the precise choice of truncation degree and whether it depends on the codimension or the Schubert class; a uniform rule should be stated explicitly.
  2. In the description of the larger family projecting onto Grassmannians, specify the base and the projection map more formally, including any flatness or smoothness statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that requires clarification. We address the major comment below and will revise the manuscript to incorporate an explicit verification as requested.

read point-by-point responses

  1. Referee: The central claim equates the blow-up along an arbitrary Schubert subvariety with a member of the family whose exceptional locus is an orbit closure on a truncated wedge power. It is not clear from the setup whether a fixed truncation degree uniformly reproduces the full ideal sheaf of the Schubert variety, including higher syzygies beyond the first-order normal directions. A concrete verification for at least one non-trivial Schubert class in the Grassmannian case (e.g., via explicit ideal generators) is needed to confirm the identification holds without missing relations.

    Authors: We agree that an explicit check strengthens the presentation. In the manuscript the truncation degree is chosen so that the orbit closure under the abelian unipotent group action defines the ideal sheaf of the Schubert variety inside the ambient projective space; the group action ensures that all higher syzygies are generated by the truncated relations. Nevertheless, to address the referee's request directly we will add a new subsection (in the Grassmannian case) containing an explicit computation for the Schubert variety corresponding to the partition (2,1) in Gr(3,6). We will list the ideal generators obtained from the truncation, compare them with the known ideal of the Schubert variety, and verify that the blow-up is recovered without missing relations. This example will be computed by hand or with standard computer-algebra tools and included in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit definitions support independent geometric theorems

full rationale

The paper opens by defining truncated Grassmannians directly as orbit closures of abelian unipotent groups on degree-truncated projectivized wedge powers. It then proves, via standard algebraic geometry, that these objects arise as exceptional loci in blow-ups of flag varieties (worked out explicitly for Grassmannians) and that the resulting varieties belong to a larger family whose fibers are spaces of collineations. No equation or claim reduces a derived object to a fitted parameter, a self-citation, or a renamed input; the derivation chain consists of constructions and theorems whose validity can be checked against the given definitions without circular reference. The central identification is therefore a genuine result rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the new definition of truncated Grassmannians together with standard facts from algebraic geometry about flag varieties, Schubert subvarieties, and blow-ups.

axioms (1)
  • domain assumption Standard properties of Grassmannians, flag varieties, and Schubert subvarieties hold as in classical algebraic geometry.
    The paper invokes these to define the ambient spaces and the blow-up operation.
invented entities (1)
  • Truncated Grassmannians no independent evidence
    purpose: To serve as the geometric objects that describe blow-ups along Schubert varieties.
    Defined in the paper as closures of orbits of abelian unipotent groups acting on degree truncations of projectivized wedge powers.

pith-pipeline@v0.9.0 · 5377 in / 1327 out tokens · 41227 ms · 2026-05-13T22:19:08.421821+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Truncated Grassmannians are defined as closures of orbits of abelian unipotent groups acting on the degree truncations of projectivized wedge powers... Theorem A. The blow-up Bl_{S_r} Gr_d(V) is isomorphic to the closure of the graph of the birational map X_{r-1} → Gr_d(V).

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We define the truncated flag variety X_σ(λ) ⊂ P(L_σ(λ)) as the closure of B^−[v_σ(λ)]. Theorem C. The blow-up of G/B along the Schubert variety S_σ admits a closed embedding into G/B × X_σ(λ) as the closure of the B^− orbit through the product [v(λ)] × [v_σ(λ)].

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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