Pith. sign in

REVIEW 4 major objections 4 minor 35 references

Canonical blow-ups of Grassmannians I: How canonical is a Kausz compactification?

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that one torus-equivariant blow-up construction on a Grassmannian produces both the Kausz compactification and the spaces of complete collineations, with the Kausz-type compactification as the total family over the…

desk verdict Genuine new idea, but the main smoothness proof has a load-bearing gap that needs a complete covering argument before acceptance. read the letter →

arxiv 2411.10763 v2 pith:KCOK2TYI submitted 2024-11-16 math.AG

Hanlong Fang , Xian Wu This is my paper · ORCID
classification math.AG MSC 14M1514L3014E0514M27
keywords KauszcompactificationcompletecollineationsGrassmannianstorusactionHilbertquotientBialynicki-BiruladecompositionLandsberg-Manivelbirationalmapssimplenormalcrossingdivisors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to establish that two classical compactification theories—the Kausz compactification of the general linear group and the spaces of complete collineations of linear maps—are two halves of one canonical operation: blow up a Grassmannian along the stable and unstable strata of a $\mathbb G_m$-action induced by a splitting $E=E_1\oplus E_2$. The resulting schemes $T_{s,p,n}$ are smooth and projective, with a boundary of $2r$ simple-normal-crossing divisors, and the two end divisors are isomorphic to $M_{s,p,n}$, the generalized space of complete collineations. Over an algebraically closed field, $M_{s,p,n}$ is the Hilbert quotient $G(p,n)/\!/\mathbb G_m$ and $T_{s,p,n}$ is its flat total family. In the square case $n=2s=2p$, $T_{p,p,2p}$ is exactly Kausz's compactification, so the paper gives a uniform explanation of why Kausz's iterative blow-ups are the natural resolution for this torus action. The same construction resolves the Landsberg-Manivel birational maps from projective spaces to Grassmannians, turning them into morphisms.

What carries the argument

The load-bearing object is the Kausz-type compactification $T_{s,p,n}$, the closure of the graph of the rational map from $G(p,n)$ to a product of Plücker projective spaces. The mechanism carrying the proofs is the Mille Crêpes coordinate atlas: coordinate charts built by iterated rank-one Gaussian elimination adapted to the Grassmannian's Plücker coordinates, on which the $\mathbb G_m$-action has the normal form $b\mapsto tb$, $a\mapsto t^{-1}a$. In these coordinates the Bialynicki-Birula decomposition is explicit: fixed loci are subschemes $D_{(p-k,k)}$, the boundary divisors $D^\pm_i$ become coordinate hyperplanes, the retraction $P_{s,p,n}$ is projection onto the invariant coordinates, and the fibers are chains of $\mathbb G_m$-stable rational curves of Plücker degree $r$. The same charts let the paper read off the isomorphism between source and sink and identify the Kausz blow-up with the resolution of the Landsberg-Manivel maps.

What would settle it

Carry out the omitted covering argument for a concrete small case, such as $G(3,6)$ with $s=3$ and $l=1$: write the equations of the union of the Mille Crêpes charts inside $R^{-1}_{s,p,n}(U_1)$ and check whether some closed point lies outside every chart. An uncovered point would falsify the smoothness and simple-normal-crossing conclusions; finding none would supply the missing step.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the Kausz-type compactification $T_{s,p,n}$—the closure of the graph of the rational map $K_{s,p,n}:G(p,n)\dashrightarrow \mathbb P(\wedge^p E)\times\prod_{k=0}^r \mathbb P(\wedge^k E_1\otimes\wedge^{n-k}E_2)$—is the canonical total space attached to the $\mathbb G_m$-action (7): its source and sink divisors $D^-_1$ and $D^+_1$ are each isomorphic to $M_{s,p,n}$, and the flat projection $P_{s,p,n}$ makes $T_{s,p,n}$ the total family over the Hilbert quotient $G(p,n)/\!/\mathbb G_m$, which is $M_{s,p,n}$ after base change to an algebraically closed field. For $n=2s=2p$, $T_{p,p,2p}$ is isomorphic to Kausz's compactification of $\mathrm{GL}_p$, so the classical Kausz compactification and the classical spaces of complete collineations are two faces of the same blow-up. The same comparison identifies the Kausz blow-up with the resolution of the Landsberg-Manivel birational maps, and the source/sink isomorphism yields explicit birational maps between projective bundles over Grassmannians.

Load-bearing premise

The load-bearing premise is the paper's claim in §3.2 that the Mille Crêpes coordinate charts cover every affine piece $R^{-1}_{s,p,n}(U_l)$, asserted by 'similarly to Lemma 3.3' without a proof; on that coverage rest the smoothness of $T_{s,p,n}$, the simple-normal-crossing divisor structure, and Theorem 1.2.

Editorial extensions

If this is right

  • Kausz's compactification of $\mathrm{GL}_p$ is a toroidal embedding of a general linear group, inheriting the divisor structure of Theorem 1.2.
  • The spaces of complete collineations of $p\times(n-p)$ matrices appear as a divisor in $T_{p,p,n}$, and the general $T_{s,p,n}$ are obtained from the Kausz compactification by iterated locally trivial fibrations over Grassmannians.
  • Over an algebraically closed field, $M_{s,p,n}$ is the Hilbert quotient $G(p,n)/\!/\mathbb G_m$, and $P_{s,p,n}:T_{s,p,n}\to M_{s,p,n}$ is its flat total family, with general fiber a chain of $\mathbb G_m$-stable rational curves of Plücker degree $r$.
  • The blow-up $KA:T_{p,p,2p}\to\mathbb P^{p^2}$ resolves the Landsberg-Manivel birational map $\mathbb P^{p^2}\dashrightarrow G(p,2p)$, and the same comparison handles the general rational maps from projective spaces to Grassmannians.
  • The isomorphism between the source and sink divisors gives explicit birational isomorphisms between the projectivized normal bundles $\mathbb P(N_1)$ and $\mathbb P(N_2)$ of two Grassmannian subvarieties.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the paper works under the standing assumption $2p\le n\le 2s$, but the coordinate construction is symmetric under exchanging the two summands of $E$ and under $p\leftrightarrow n-p$; if the missing covering proof in §3.2 is supplied, the same smoothness and Hilbert-quotient statements should extend to the full parameter range without new geometry.
  • Editorial inference: because the fibers of $P_{s,p,n}$ are explicit chains of rational curves in the Mille Crêpes charts, one can use $T_{s,p,n}$ as a computational bridge: intersection numbers on $M_{s,p,n}$ could be pushed forward from a smooth variety whose boundary divisors are coordinate hyperplanes.
  • Editorial inference: the paper establishes the construction but not a uniqueness theorem; a natural test of 'canonical' would be whether every smooth $\mathbb G_m$-equivariant compactification of the general linear group with source and sink isomorphic and resolving the Euler foliation must coincide with $T_{s,p,n}$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper constructs schemes T_{s,p,n} and M_{s,p,n} as closures of a rational map from the Grassmannian G(p,n) to a product of projective spaces, and realizes T_{s,p,n} as an iterated blow-up of G(p,n) according to the torus action (7). The main theorems assert that T_{s,p,n} is smooth and projective over Spec Z, that its boundary consists of 2r smooth divisors with simple normal crossings, that there is a flat retraction onto M_{s,p,n}, that M_{s,p,n} is wonderful, and that M_{s,p,n} is isomorphic to the Hilbert quotient G(p,n)//G_m. The paper also identifies T_{p,p,2p} with the Kausz compactification of GL_p, derives toroidal embeddings and resolutions of the Landsberg-Manivel birational maps, and gives fibrations of these spaces over smaller Grassmannians.

Significance. If the missing proof of the atlas covering is supplied, the paper would be a substantial and useful unification: it would give a single blow-up picture for complete collineations and Kausz compactifications, identify the total family over the Hilbert quotient, and provide concrete coordinate charts that should be checkable in examples. The explicit coordinate constructions and worked examples (Examples 3.1 and 3.5) are valuable, and the claimed divisor equations in Lemma 4.3 are concrete and testable. The main geometric claims are not circular and do not rely on fitted parameters, but several load-bearing assertions are currently stated without proof.

major comments (4)
  1. [§3.2] The assertion after the definition of J^τ_l that ∪_{τ∈J_l} A_τ = R^{-1}_{s,p,n}(U_l) is the only proof that the Mille Crêpes charts form an atlas, but it is not proved: the text says 'Similarly to Lemma 3.3, we can further prove' and then proceeds. Lemma 3.3 itself is proved by an induction over auxiliary charts A^τ_k on intermediate blow-ups Y^p_k, and the general l-case contains two Gaussian-elimination blocks, so the reduction is not automatic. This coverage statement is used in Proposition 3.6 for smoothness of T_{s,p,n}, in Lemma 4.3 for the simple normal crossing divisor structure, and in Theorem 1.2(C) for the orbit stratification; if some point of R^{-1}_{s,p,n}(U_l) lies outside every A_τ, the main theorem is not supported. Please provide the complete proof, or a rigorous computational verification of the covering for all 0≤l≤r and all parameter ranges. The companion statement that J^τ_l extends to an embedding is also left as 'Similarly to Lemma 3.2' and should be written out.
  2. [§2] The statement 'Without loss of generality, we assume that 2p ≤ n ≤ 2s' is not justified. The quantities r, the charts U_l, and the blow-up centers depend on s,p,n, and no symmetry is stated that reduces the general case to this range. Since Theorem 1.2 is asserted for all 0<p<n, either the reduction should be proved explicitly (for instance by a duality or transpose argument) or the statement of the theorem should be restricted to the range 2p≤n≤2s.
  3. [§5.1] In Proposition 5.1, the claim that K_{n-p,p,n}∘LM∘KA extends to an embedding is central to the identification KA_{p,n} ≅ T_{n-p,p,n} and hence to Theorem 1.3 and Corollary 1.8, but the proof is only 'Similar to the proof of Lemma 3.2'. The coordinate systems used for KA_{p,n} in (35) and the Mille Crêpes charts of §3 are not identical, and the required cancellation and injectivity computations are not shown. This step should be written out in detail or replaced by a precise reference to a proved statement that applies directly.
  4. [§4] Property (D) of Theorem 1.2 asserts that D^-_1 is wonderful. The proof says 'as in the case of the spaces of complete collineations ... we can show' without supplying the orbit/stratum correspondence or verifying the defining properties of a wonderful variety on D^-_1. In particular, one needs to check that the boundary divisors ˇD_i are smooth with simple normal crossings and that the closures of G-orbits in D^-_1 match the intersections of the ˇD_i. This is a main assertion of the paper and should be proved at the same level of detail as the other parts of Theorem 1.2.
minor comments (4)
  1. [§3.2] In the definition of J_l, 'paritial permutation' is a typo for 'partial permutation'.
  2. [§4] In the proof of Lemma 4.12, the phrase 'the restriction of ϕ to Z_q' uses an undefined morphism ϕ; it should presumably be R_{s,p,n}.
  3. [§4] In the proof of Theorem 1.7, 'Property (c) in Lemma 1.7' should read 'Lemma 4.12(c)'; the reference to Lemma 1.7 does not match the numbering.
  4. [§3.2] In Example 3.5, the notation C^{16} is used for the affine scheme of coordinates over Spec Z; using A^{16} would be consistent with the rest of the paper.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central derivation is independent, with one non-load-bearing self-reference and one unproved atlas-coverage proof gap flagged for referees.

full rationale

The paper's central objects are defined directly as closures of an explicit Plücker-coordinate rational map and then independently realized as iterated blow-ups of G(p,n) along determinantal subschemes (Lemma 2.2). The main claims about smoothness, simple normal crossing divisors, the Białynicki-Birula decomposition, the flat retraction, and the Hilbert quotient identification are argued from these coordinate constructions together with external theorems by Białynicki-Birula, Thaddeus, Kollár, Fulton–Pandharipande, and Occhetta et al., none of which are authored by the present authors. The only self-reference is the abstract's disclosure that parts of the paper are taken from the first author's earlier arXiv post; this is provenance, not a proof input, so it is not load-bearing. One genuine proof gap, though not a circularity, appears in §3.2: the text says 'Similarly to Lemma 3.3, we can further prove that ∪_{τ∈J_l} A_τ = R^{-1}_{s,p,n}(U_l)' for every l, but the proof is omitted. Since Proposition 3.6, Lemma 4.3, and Theorem 1.2 rely on this atlas covering, a referee should require a complete proof of that statement. This is an omitted justification, not a circular reduction: no definition is framed in terms of the target isomorphism, and no parameter is fitted to force the conclusions. The verdict is therefore no significant circularity; the score of 1 reflects only the minor, non-load-bearing self-reference in the abstract.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central contribution is a construction (T_{s,p,n}) rather than a postulate; no new particles or dimensions. The paper's axioms are standard algebraic geometry tools plus the unproved WLOG range reduction.

assumptions (4)
  • standard math Bialynicki-Birula decomposition and fixed-point scheme properties for smooth projective schemes with torus actions
    Used throughout §4 to define D^±_k and compute fixed-point components; cited to Bialynicki-Birula [2].
  • domain assumption Reduction to 2p <= n <= 2s is legitimate without loss of generality
    Stated in §2 without proof; the paper's theorems are proven only under this range. If the implied symmetries p -> n-p and s -> n-s fail, the results do not cover all p, s, n.
  • standard math S_k is the scheme-theoretic union of V^+_{(p-k-1,k+1)} and V^-_{(p-k+1,k-1)}
    Lemma 4.2, cited to Bruns-Vetter [6, Theorem 1.4]. Load-bearing for identifying the divisor structure.
  • standard math Smoothness of the Hilbert quotient G(p,n)//G_m via Occhetta-Romano-Conde-Wisniewski [24, Proposition 6.3] and vanishing of H^1(C, T G(p,n)|_C)
    Used in the proof of Theorem 1.7; the vanishing is standard because T G(p,n) is globally generated, but the smoothness transfer is cited rather than re-derived.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Canonical blow-ups of Grassmannians I: How canonical is a Kausz compactification?." pith.science (2026). https://pith.science/paper/KCOK2TYI

@misc{pith2026241110763,
  author       = {Pith},
  title        = {Pith review of: Canonical blow-ups of Grassmannians I: How canonical is a Kausz compactification?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCOK2TYI}},
  note         = {Machine review of arXiv:2411.10763}
}
abstract

In this paper, we develop a simple uniform picture incorporating the Kausz compactifications and the spaces of complete collineations by blowing up Grassmannians $G(p,n)$ according to a torus action $\mathbb G_m$. We show that each space of complete collineations is isomorphic to any maximal-dimensional connected component of the $\mathbb G_m$-fixed point scheme of a Kausz-type compactification. We prove that the Kausz-type compactification is the total family over the Hilbert quotient $G(p,n)/ \! \! / \mathbb G_m$ which is isomorphic to the space of complete collineations. In particular, the Kausz compactifications are toroidal embeddings of general linear groups in the sense of Brion-Kumar. We also show that the Kausz-type compactifications resolve the Landsberg-Manivel birational maps from projective spaces to Grassmannians, by comparing Kausz's construction with ours. As an application, by studying the foliation we derive resolutions of certain birational maps among projective bundles over Grassmannians. The results in this paper are partially taken from the first author's earlier arxiv post (Canonical blow-ups of grassmann manifolds, arxiv:2007.06200), which has been revised and expanded in collaboration with the second author.

Figures

Figures reproduced from arXiv: 2411.10763 by the authors.

Figure 1
Figure 1. The foliation on G(2, 4). D + 1 is the exceptional divisor over the infinity point V0,2. D − 1 is the exceptional divisor over the origin V2,0. D + 2 is the exceptional divisor over the union of all orbital rational curves between V1,1 and V0,2; D − 2 is the exceptional divisor over the union of all orbital rational curves between V1,1 and V2,0. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 34 canonical work pages

  1. [1]

    Alguneid

    A. Alguneid. Degeneration of space collineations. Proc. Egyptian Acad. Sci., 7:1–17 (1952), 1951

    work page 1952

  2. [2]

    Bia lynicki-Birula

    A. Bia lynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2), 98:480– 497, 1973

    work page 1973

  3. [3]

    Bia lynicki-Birula and A

    A. Bia lynicki-Birula and A. Sommese. Quotients by C∗ and SL(2 , C) actions. Trans. Amer. Math. Soc., 279(2):773–800, 1983

    work page 1983

  4. [4]

    Bravi and G

    P. Bravi and G. Pezzini. Primitive wonderful varieties. Math. Z., 282(3-4):1067–1096, 2016

    work page 2016

  5. [5]

    Brion and S

    M. Brion and S. Kumar. Frobenius splitting methods in geometry and representationtheory, volume 231 of Progress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 2005

    work page 2005

  6. [6]

    Bruns and U

    W. Bruns and U. Vetter. Determinantal rings, volume 1327 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988

    work page 1988

  7. [7]

    De Concini and C

    C. De Concini and C. Procesi. Complete symmetric varieties. In Invariant theory (Montecatini, 1982), volume 996, pages 1–44. Springer, Berlin, 1983

    work page 1982

  8. [8]

    De Concini and C

    C. De Concini and C. Procesi. Complete symmetric varieties. II. Intersection theory. In Adv. Stud. Pure Math. Vol. 6, 481–513, North-Holland, Amsterdam, 1985

    work page 1985

Show all 35 references
  1. [9]

    C. Ding. Birational transformations on irreducible compact Hermitian symmetric spaces. Int. Math. Res. Not. IMRN, (11):9266–9291, 2024

    2024

  2. [10]

    Faltings

    G. Faltings. Explicit resolution of local singularities of moduli-spaces. J. Reine Angew. Math., 483:183–196, 1997

    1997

  3. [11]

    A. Fujiki. Fixed points of the actions on compact K¨ ahler manifolds. Publ. Res. Inst. Math. Sci., 15(3):797–826, 1979

    1979

  4. [12]

    Fulton and R

    W. Fulton and R. Pandharipande. Notes on stable maps and quantum cohomology. volume 62 of Proc. Sympos. Pure Math., pages 45–96. Amer. Math. Soc., Providence, RI, 1997

    1997

  5. [13]

    X. He. Normality and Cohen-Macaulayness of local models of Shimura varieties. Duke Math. J., 162(13):2509–2523, 2013

    2013

  6. [14]

    J. Hwang. Nondeformability of the complex hyperquadric. Invent. Math., 120(2):317–338, 1995

    1995

  7. [15]

    M. M. Kapranov. Chow quotients of Grassmannians. I. volume 16 of Adv. Soviet Math., pages 29–110. Amer. Math. Soc., Providence, RI, 1993

    1993

  8. [16]

    I. Kausz. A modular compactification of the general linear group. Doc. Math., 5:553–594, 2000

    2000

  9. [17]

    F. Knop. The Luna-Vust theory of spherical embeddings. In Proceedings of the Hyderabad Conference on Algebraic Groups, pages 225–249. Manoj Prakashan, Madras, 1991

    1991

  10. [18]

    Springer-Verlag, Berlin, 1996

    J´ anos Koll´ ar.Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1996

    1996

  11. [19]

    D. Laksov. The geometry of complete linear maps. Ark. Mat., 26(2):231–263, 1988

    1988

  12. [20]

    Laksov, A

    D. Laksov, A. Lascoux, and A. Thorup. On Giambelli’s theorem on complete correlations. Acta Math., 162(3-4):143–199, 1989. 26

    1989

  13. [21]

    Landsberg and L

    J. Landsberg and L. Manivel. Construction and classification of complex simple Lie algebras via projective geometry. Selecta Math. (N.S.), 8(1):137–159, 2002

    2002

  14. [22]

    D. Luna. Toute vari´ et´ e magnifique est sph´ erique.Transform. Groups, 1(3):249–258, 1996

    1996

  15. [23]

    G. Lusztig. Parabolic character sheaves. I. Mosc. Math. J., 4(1):153–179, 311, 2004

    2004

  16. [24]

    Occhetta, E

    G. Occhetta, E. Romano, L. Conde, and J. Wi´ sniewski. Chow quotients of C∗-action. arxiv:2310.18623, 2023

    2023 arXiv

  17. [25]

    G. Pezzini. Lectures on wonderful varieties. Acta Math. Sin. (Engl. Ser.), 34(3):417–438, 2018

    2018

  18. [26]

    J. Semple. The variety whose points represent complete collineations of Sr on S′ r. Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5), 10:201–208, 1951

    1951

  19. [27]

    F. Severi. I fondamenti della geometria numerativa. Ann. Mat. Pura Appl. (4), 19:153–242, 1940

    1940

  20. [28]

    Y. Siu. Nondeformability of the complex projective space. J. Reine Angew. Math., 399:208–219, 1989

    1989

  21. [29]

    T. A. Springer. Intersection cohomology of B × B-orbit closures in group compactifications. volume 258, pages 71–111. 2002. With an appendix by Wilberd van der Kallen, Special issue in celebration of Claudio Procesi’s 60th birthday

    2002

  22. [30]

    E. Study. Ueber die Geometrie der Kegelschnitte, insbesondere deren Charakteristikenproblem. Math. Ann., 27(1):58–101, 1886

  23. [31]

    Thaddeus

    M. Thaddeus. Complete collineations revisited. Math. Ann., 315(3):469–495, 1999

    1999

  24. [32]

    Thorup and S

    A. Thorup and S. Kleiman. Complete bilinear forms. In Algebraic geometry (Sundance, UT, 1986), volume 1311 of Lecture Notes in Math., pages 253–320. Springer, Berlin, 1988

    1986

  25. [33]

    J. A. Tyrrell. Complete quadrics and collineations in Sn. Mathematika, 3:69–79, 1956

    1956

  26. [34]

    Vainsencher

    I. Vainsencher. Complete collineations and blowing up determinantal ideals. Math. Ann., 267(3):417–432, 1984

    1984

  27. [35]

    B. L. van der Waerden. Zur algebraischen Geometrie. XV. Math. Ann., 115(1):645–655, 1938. Hanlong Fang, School of Mathematical Sciences, Peking University, Beijing, 100871, China. E-mail address: hlfang@pku.edu.cn Xian Wu, School of Mathematical Sciences, Peking University, Be...

    1938

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.