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arxiv: 2604.25645 · v1 · submitted 2026-04-28 · 🧮 math.AG

GIT quotient of minimal dimensional Schubert variety modulo a subtorus

Arkadev Ghosh , S. S. Kannan This is my paper

Pith reviewed 2026-05-07 15:22 UTC · model grok-4.3

classification 🧮 math.AG
keywords GIT quotientSchubert varietyGrassmannianiterated projective bundlesubtorus actionsemistable pointsfundamental weight
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The pith

The GIT quotient of the minimal Schubert variety X(w_{r,n}) by its peak subtorus T_{J_r} is the total space of the r-th stage of an iterated projective space bundle over P^{q-1}.

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

The paper proves that when n = r q + 1 for integer q at least 2, the geometric invariant theory quotient of the Schubert variety X(w_{r,n}) inside the Grassmannian G_{r,n} by the subtorus T_{J_r} is isomorphic to the total space of the r-th stage of an iterated projective space bundle over projective space of dimension q minus 1. This uses the earlier identification of X(w_{r,n}) as the unique minimal-dimensional Schubert variety that admits semistable points for the T-linearized ample line bundle associated to n times the r-th fundamental weight. A reader would care because the result replaces an abstract quotient by an explicit bundle space whose geometry is easier to analyze directly. The construction relies on the peaks of the Weyl group element w_{r,n} to select the subtorus whose action reduces the quotient to the claimed iterated bundle.

Core claim

When n equals r q plus one with q at least 2, the GIT quotient of X(w_{r,n}) modulo T_{J_r} is isomorphic to the total space of the r-th stage of an iterated projective space bundle over projective space of dimension q minus 1.

What carries the argument

The subtorus T_{J_r} generated by the one-parameter subgroups of the maximal torus that correspond to the peaks of w_{r,n}, acting on the Schubert variety X(w_{r,n}) via the linearized ample line bundle L(n omega_r).

If this is right

  • The quotient space carries a natural projection to P^{q-1} whose fibers are iterated projective spaces of total stage r.
  • Semistable orbits correspond to points in this explicit bundle total space.
  • The isomorphism preserves the algebraic structure needed to compute T_{J_r}-invariants on the Schubert variety.

Where Pith is reading between the lines

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

  • The bundle description may make it feasible to compute the cohomology ring of the quotient by iterating the projective bundle formula.
  • Similar subtorus quotients could be examined for other Schubert varieties or for Grassmannians with different fundamental weights.
  • One could check whether the same iterated-bundle form appears when the ambient dimension n deviates slightly from r q plus one.

Load-bearing premise

The numerical condition that n equals r q plus one with q at least 2, together with the uniqueness of X(w_{r,n}) as the minimal Schubert variety admitting semistable points for the given linearized line bundle.

What would settle it

An explicit point in the GIT quotient whose local structure or orbit dimension fails to match the fiber of the r-th stage iterated projective space bundle over P^{q-1}, or a mismatch in the Picard group or canonical class between the two spaces.

read the original abstract

Let $G=PSL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $\omega_{r}$ denote the $r^{th}$ fundamental weight. Let $\mathcal{L}(n\omega_{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $n\omega_{r}$ of $T$. In an earlier work of Kannan and Sardar, it is proved that there is a unique minimal dimensional Schubert variety $X(w_{r,n})$ in $G_{r,n}$ admitting semistable points for the $T$-linearized ample line bundle $\mathcal{L}(n\omega_{r})$. Assume that $n=rq+1$, where $r,q\in\mathbb{N}$ and $q\geq 2$. In this paper, we study the GIT quotient of $X(w_{r,n})$ modulo a subtorus $T_{J_{r}}$ of $T$ generated by the one parameter subgroups of $T$ corresponding to the peaks of $w_{r,n}$. We prove that the GIT quotient of $X(w_{r,n})$ modulo $T_{J_{r}}$ is isomorphic to the total space of the $r^{th}$ stage of an iterated projective space bundle over $\mathbb{P}^{q-1}$.

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

0 major / 2 minor

Summary. The paper proves that, assuming n = rq + 1 with q ≥ 2, the GIT quotient of the minimal-dimensional Schubert variety X(w_{r,n}) in G_{r,n} by the subtorus T_{J_r} (generated by one-parameter subgroups corresponding to the peaks of w_{r,n}) is isomorphic to the total space of the r-th stage of an iterated projective space bundle over P^{q-1}. The argument describes the semistable locus for the subtorus action, constructs an explicit isomorphism, and verifies the necessary properties of the ring of invariants, building on the prior Kannan–Sardar result that X(w_{r,n}) is the unique minimal-dimensional Schubert variety admitting semistable points for the T-linearized bundle L(n ω_r).

Significance. If the result holds, the explicit isomorphism provides a concrete geometric model for this GIT quotient, linking Schubert varieties to iterated projective bundles. This strengthens the study of semistable loci and quotients in Grassmannians and may enable further computations of invariants or cohomology. The construction via semistable loci and ring-of-invariants verification is a positive feature.

minor comments (2)
  1. §1 (Introduction): the definition of the subtorus T_{J_r} and the peaks of w_{r,n} would benefit from a brief self-contained description or explicit reference to the relevant section in Kannan–Sardar, to improve accessibility.
  2. The section constructing the iterated bundle (likely §4 or §5): the inductive description of the r stages could include a small diagram or explicit coordinate description for the base case r=1 to clarify the iteration.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and for recommending minor revision. No specific major comments were raised in the report, so we have no individual points to address. We will incorporate any minor editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper cites prior work by Kannan and Sardar (with author overlap) solely to identify the unique minimal Schubert variety X(w_{r,n}) admitting semistable points under the T-linearized bundle L(n ω_r). This is used as an explicit hypothesis together with the assumption n = rq + 1. The central result—the isomorphism of the GIT quotient by the subtorus T_{J_r} to the total space of the r-th iterated projective bundle over P^{q-1}—is established by direct description of the semistable locus, explicit invariants, and bundle construction. No step reduces the claimed isomorphism to the cited fact by definition, renaming, or construction; the derivation remains self-contained and independent of the supporting citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior existence result for the minimal Schubert variety and standard background facts from algebraic geometry. No free parameters are fitted to data, and no new entities are postulated in the abstract.

axioms (2)
  • domain assumption There exists a unique minimal dimensional Schubert variety X(w_{r,n}) in G_{r,n} admitting semistable points for the T-linearized ample line bundle L(n ω_r).
    Cited as proved in earlier work of Kannan and Sardar.
  • standard math Standard properties of GIT quotients, torus actions, Schubert varieties, and line bundles on Grassmannians hold as background.
    These are implicit throughout the setup involving G=PSL(n,C), T, and the subtorus T_{J_r}.

pith-pipeline@v0.9.0 · 5538 in / 1637 out tokens · 94692 ms · 2026-05-07T15:22:57.376973+00:00 · methodology

discussion (0)

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Reference graph

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