Pith. sign in

REVIEW 1 major objections 2 minor 53 references

Higher-dimensional Losev-Manin spaces are fibrations over products of projective spaces with classical Losev-Manin fibers and normalizations of Chow quotients.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-05-24 06:49 UTC

load-bearing objection The paper gives a fibration structure and Chow quotient identification for higher-dimensional Losev-Manin spaces, plus a blow-up criterion that shows the Chen-Gibney-Krashen spaces are not Mori dream spaces for n at least 9 in any dimension. the 1 major comments →

arxiv 2308.07911 v2 submitted 2023-08-15 math.AG

Higher-dimensional Losev-Manin spaces and their geometry

Patricio Gallardo , Javier Gonz\'alez-Anaya , Jos\'e Luis Gonz\'alez , Evangelos Routis This is my paper
classification math.AG
keywords Losev-Manin spacestoric compactificationsMori dream spacesChow quotientsmoduli of pointsblow-upsalgebraic geometry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper examines toric compactifications of the moduli space of n distinct labeled points in affine d-space modulo translation and scaling. It establishes that these higher-dimensional Losev-Manin spaces form a fibration over a product of projective spaces, with each fiber isomorphic to the classical Losev-Manin space. The spaces are also isomorphic to the normalization of a Chow quotient. The authors introduce a criterion that determines when the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. They apply the criterion to prove that the Chen-Gibney-Krashen generalization is not a Mori dream space for n at least nine, independent of dimension.

Core claim

The higher-dimensional Losev-Manin spaces compactify the moduli space of n distinct labeled points in affine space modulo translation and scaling. These spaces are a fibration over a product of projective spaces with fibers isomorphic to the Losev-Manin space and are isomorphic to the normalization of a Chow quotient. A criterion decides whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space; as an application the Chen-Gibney-Krashen generalization is not a Mori dream space for n at least nine, independent of dimension.

What carries the argument

The fibration structure of the higher-dimensional Losev-Manin spaces together with the blow-up criterion along subtorus closures that tests the Mori dream space property.

Load-bearing premise

The higher-dimensional moduli spaces of labeled points admit toric compactifications whose geometry is governed by the same combinatorial data that controls the classical Losev-Manin spaces.

What would settle it

An explicit computation of the Mori dream space property for the Chen-Gibney-Krashen space with exactly nine points in dimension two, showing it satisfies the property, would falsify the application of the criterion.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The geometry of the higher-dimensional spaces reduces to the one-dimensional Losev-Manin case via the fibration.
  • The normalization of the Chow quotient supplies an explicit toric model for these moduli spaces.
  • The blow-up criterion classifies additional toric blow-ups as Mori dream spaces or not.
  • The Chen-Gibney-Krashen generalization fails to be a Mori dream space whenever the number of points reaches nine or more.

Where Pith is reading between the lines

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

  • Similar fibration structures may appear in other moduli spaces of point configurations in affine space.
  • The criterion offers a practical test for the Mori dream property in further toric blow-up constructions.
  • Connections to GIT or other quotient constructions could be explored using the normalization statement.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper introduces higher-dimensional toric compactifications of the moduli space of n labeled points in affine d-space modulo translation and scaling, called higher-dimensional Losev-Manin spaces. It proves these spaces form a fibration over a product of projective spaces with fibers isomorphic to the classical Losev-Manin space and are isomorphic to the normalization of a Chow quotient. It also states a criterion deciding when the blow-up of a toric variety along the closure of a subtorus is a Mori dream space, and applies the criterion to show that the Chen-Gibney-Krashen generalization is not a Mori dream space for n≥9 independently of dimension.

Significance. If the central identifications and the MDS criterion hold, the work provides a useful geometric extension of Losev-Manin spaces with explicit fibration and Chow-quotient descriptions, plus a combinatorial tool for detecting Mori dream spaces that yields a uniform non-MDS result across dimensions. The dimension-independent application is a concrete strength if the criterion's combinatorial conditions are shown to be insensitive to ambient dimension.

major comments (1)
  1. [Criterion section] Criterion section (likely §4 or §5): the stated criterion for the blow-up of a toric variety X along the closure of a subtorus to be MDS is given in terms of fan data, weights, and class-group generators, but the manuscript does not explicitly verify that the finite-generation or movable-cone polyhedrality conditions remain unchanged when the ambient dimension d increases (additional rays or relations in Cl(X) appear for d>1). This is load-bearing for the application claiming the Chen-Gibney-Krashen spaces fail to be MDS for n≥9 independently of dimension.
minor comments (2)
  1. [Introduction] Notation for the higher-dimensional moduli space (e.g., the precise definition of the toric fan in terms of the labeled points) could be clarified with an explicit example for small n and d=2 to aid readability.
  2. [Abstract] The abstract states the fibration and Chow-quotient results without indicating the section where the proofs appear; adding section references would improve navigation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for an explicit verification of dimension-independence in the criterion. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Criterion section] Criterion section (likely §4 or §5): the stated criterion for the blow-up of a toric variety X along the closure of a subtorus to be MDS is given in terms of fan data, weights, and class-group generators, but the manuscript does not explicitly verify that the finite-generation or movable-cone polyhedrality conditions remain unchanged when the ambient dimension d increases (additional rays or relations in Cl(X) appear for d>1). This is load-bearing for the application claiming the Chen-Gibney-Krashen spaces fail to be MDS for n≥9 independently of dimension.

    Authors: The criterion is stated combinatorially in terms of the fan of X, the weights defining the subtorus, and a choice of generators for Cl(X). For the Chen-Gibney-Krashen spaces the fan and the relevant class-group generators are constructed uniformly in d, so the finite-generation and polyhedrality conditions used to detect non-MDS behavior depend only on those combinatorial data and are insensitive to additional rays that appear for d>1. Nevertheless, we agree that an explicit verification paragraph is missing. In the revised version we will insert a short subsection (or remark) immediately after the statement of the criterion that confirms the conditions remain unchanged with d, thereby making the dimension-independent application fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations are independent geometric identifications

full rationale

The paper derives that higher-dimensional Losev-Manin spaces are fibrations over products of projective spaces with classical Losev-Manin fibers and are normalizations of Chow quotients, plus a toric blow-up criterion for Mori dream spaces applied to the Chen-Gibney-Krashen spaces. These steps rely on standard toric geometry and moduli identifications rather than self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that reduce claims to tautologies. The abstract and description provide no quoted equations or reductions where outputs equal inputs by construction, so the derivation chain is self-contained against external combinatorial and geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are mentioned; the work rests on the standard axioms of algebraic geometry and toric varieties.

pith-pipeline@v0.9.0 · 5702 in / 1291 out tokens · 21161 ms · 2026-05-24T06:49:31.998833+00:00 · methodology

0 comments
read the original abstract

The classical Losev-Manin space is a toric compactification of the moduli space of $n$ points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of $n$ distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces -- with fibers isomorphic to the Losev-Manin space -- and that they are isomorphic to the normalization of a Chow quotient. Moreover, we present a criterion to decide whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. As an application, we demonstrate that a related generalization of the moduli space of pointed rational curves constructed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    144, Cambridge University Press, 2015

    Ivan Arzhantsev, Ulrich Derenthal, J \"u rgen Hausen, and Antonio Laface, Cox rings, no. 144, Cambridge University Press, 2015

    work page 2015

  2. [2]

    799, 1--56

    Valery Alexeev, Philip Engel, and Alan Thompson, Stable pair compactification of moduli of K 3 surfaces of degree 2 , Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) 2023 (2023), no. 799, 1--56

    work page 2023

  3. [3]

    Valery Alexeev, Complete moduli in the presence of semiabelian group action, Annals of Mathematics (2002), 611--708

    work page 2002

  4. [4]

    3, 296--319

    Kenneth Ascher and Samouil Molcho, Logarithmic stable toric varieties and their moduli, Algebraic Geometry 3 (2016), no. 3, 296--319

    work page 2016

  5. [5]

    Birkar, Paolo Cascini, Christopher Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J

    Caucher. Birkar, Paolo Cascini, Christopher Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405--490

    work page 2010

  6. [6]

    1, 277--301

    Lev Borisov and Zheng Hua, On the conjecture of King for smooth toric Deligne--Mumford stacks , Advances in Mathematics 221 (2009), no. 1, 277--301

    work page 2009

  7. [7]

    3, 425--432

    Laura Costa, Sandra Di Rocco, and Rosa Maria Miro-Roig, Derived category of fibrations, Mathematical Research Letters 18 (2011), no. 3, 425--432

    work page 2011

  8. [8]

    Algebraic Geom

    Linda Chen, Angela Gibney, and Daniel Krashen, Pointed trees of projective spaces, J. Algebraic Geom. 18 (2009), no. 3, 477--509

    work page 2009

  9. [9]

    3, 365--382

    Olga Chuvashova, The main component of the toric Hilbert scheme , Tohoku Mathematical Journal, Second Series 60 (2008), no. 3, 365--382

    work page 2008

  10. [10]

    800, 1--44

    Ana-Maria Castravet, Antonio Laface, Jenia Tevelev, and Luca Ugaglia, Blown-up toric surfaces with non-polyhedral effective cone, Journal f\"ur die reine und angewandte Mathematik (Crelles Journal) 2023 (2023), no. 800, 1--44

    work page 2023

  11. [11]

    Alastair Craw and Diane Maclagan, Fiber fans and toric quotients, Discrete & Computational Geometry 37 (2007), 251--266

    work page 2007

  12. [12]

    9, 2313--2329

    Qile Chen and Matthew Satriano, Chow quotients of toric varieties as moduli of stable log maps , Algebra & Number Theory 7 (2013), no. 9, 2313--2329

    work page 2013

  13. [13]

    Reine Angew

    Ana-Maria Castravet and Jenia Tevelev, Hypertrees, projections, and moduli of stable rational curves, J. Reine Angew. Math. 675 (2013), 121--180

    work page 2013

  14. [14]

    , Derived category of moduli of pointed curves-i, Algebraic geometry 7 (2020), 722--757

    work page 2020

  15. [15]

    Igor Dolgachev and Benjamin Howard, Configuration spaces of complex and real spheres, Recent Advances in Algebraic Geometry, ed. C. Hacon, M. Mustata and M. Popa, London Math. Soc. Lect. Notes, Cambridge Univ. Press (2015), 156--179

    work page 2015

  16. [16]

    296, Cambridge University Press, 2003

    Igor Dolgachev, Lectures on invariant theory, vol. 296, Cambridge University Press, 2003

    work page 2003

  17. [17]

    2, 205--216

    Olivia Dumitrescu, Elisa Postinghel, and Stefano Urbinati, Cones of effective divisors on the blown-up P ^3 in general lines , Rendiconti del Circolo Matematico di Palermo Series 2 66 (2017), no. 2, 205--216

    work page 2017

  18. [18]

    4, 4119--4139

    Sandra Di Rocco and Luca Schaffler, Families of pointed toric varieties and degenerations, Mathematische Zeitschrift 301 (2022), no. 4, 4119--4139

    work page 2022

  19. [19]

    1, 183--225

    William Fulton and Robert M ac P herson, A compactification of configuration spaces, Annals of Mathematics 139 (1994), no. 1, 183--225

    work page 1994

  20. [20]

    Algebra 539 (2019), 118--137

    Javier Gonz\' a lez Anaya, Jos\' e Luis Gonz\' a lez, and Kalle Karu, Constructing non- M ori D ream S paces from negative curves , J. Algebra 539 (2019), 118--137. 3995238

    work page 2019

  21. [21]

    Pure Appl

    , On a family of negative curves, J. Pure Appl. Algebra 223 (2019), no. 11, 4871--4887. 3955045

    work page 2019

  22. [22]

    3, 1342--1362

    Javier Gonz \'a lez Anaya, Jos \'e Luis Gonz \'a lez, and Kalle Karu, Curves generating extremal rays in blowups of weighted projective planes, Journal of the London Mathematical Society 104 (2021), no. 3, 1342--1362

    work page 2021

  23. [23]

    , The geography of negative curves, arXiv preprint arXiv:2104.03950 (2021), To appear in The Michigan Mathematical Journal

    work page arXiv 2021

  24. [24]

    Javier Gonz\'alez Anaya, Jos\'e Luis Gonz\'alez, and Kalle Karu, Nonexistence of negative curves, International Mathematics Research Notices (2023), rnac355

    work page 2023

  25. [25]

    Noah Giansiracusa and William Danny Gillam, On K apranov's description of M_ 0,n as a C how quotient , Turkish J. Math. 38 (2014), no. 4, 625--648

    work page 2014

  26. [26]

    2, 457--477

    Patricio Gallardo and Noah Giansiracusa, Modular interpretation of a non-reductive Chow quotient , Proceedings of the Edinburgh Mathematical Society 61 (2018), no. 2, 457--477

    work page 2018

  27. [27]

    4, 2567--2583

    Patricio Gallardo, Jos \'e Luis Gonz \'a lez, and Evangelos Routis, The F ulton- M ac P herson compactification is not a M ori dream space , Mathematische Zeitschrift 302 (2022), no. 4, 2567--2583

    work page 2022

  28. [28]

    5, 984--996

    Jos \'e Luis Gonz \'a lez and Kalle Karu, Some non-finitely generated C ox rings , Compositio Mathematica 152 (2016), no. 5, 984--996

    work page 2016

  29. [29]

    7, 855--885

    Angela Gibney and Diane Maclagan, Equations for C how and H ilbert quotients , Algebra & Number Theory 4 (2011), no. 7, 855--885

    work page 2011

  30. [30]

    Patricio Gallardo and Evangelos Routis, Wonderful compactifications of the moduli space of points in affine and projective space, European Journal of Mathematics 3 (2017), 520--564

    work page 2017

  31. [31]

    Ulrich G \"o rtz and Torsten Wedhorn, Algebraic Geometry I: Schemes , Wiesbaden: Vieweg + Teubner, 2010, 1st Edition

    work page 2010

  32. [32]

    Noah Giansiracusa and Xian Wu, Chow quotients of Grassmannians by diagonal subtori , Facets of Algebraic Geometry: Volume 1: A Collection in Honor of William Fulton's 80th Birthday 472 (2022), 241

    work page 2022

  33. [33]

    2, 316--352

    Brendan Hassett, Moduli spaces of weighted pointed stable curves, Advances in Mathematics 173 (2003), no. 2, 316--352

    work page 2003

  34. [34]

    10, 4426--4445

    Zhuang He, Mori dream spaces and blow-ups of weighted projective spaces, Journal of Pure and Applied Algebra 223 (2019), no. 10, 4426--4445

    work page 2019

  35. [35]

    3-4, 1339--1358

    Juergen Hausen, Simon Keicher, and Antonio Laface, On blowing up the weighted projective plane, Mathematische Zeitschrift 290 (2018), no. 3-4, 1339--1358

    work page 2018

  36. [36]

    6, 1507--1521

    Lutz Hille and Markus Perling, A counterexample to king's conjecture, Compositio Mathematica 142 (2006), no. 6, 1507--1521

    work page 2006

  37. [37]

    u rgen Hausen and Hendrik S \

    J \"u rgen Hausen and Hendrik S \"u , The C ox ring of an algebraic variety with torus action , Advances in Mathematics 225 (2010), no. 2, 977--1012

    work page 2010

  38. [38]

    3, 399--440

    Yi Hu, Topological aspects of C how quotients , Journal of Differential Geometry 69 (2005), no. 3, 399--440

    work page 2005

  39. [39]

    Mikhail Kapranov, C how quotients of G rassmannians. I , I . M . G elfand S eminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29--110

    work page 1993

  40. [40]

    Alastair King, Tilting bundles on some rational surfaces, https://people.bath.ac.uk/masadk/papers/tilt.pdf, 1997, Accessed: 2023-07-24

    work page 1997

  41. [41]

    J \'a nos Koll \'a r, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996

    work page 1996

  42. [42]

    200, Cambridge University Press, 2013

    , Singularities of the minimal model program, vol. 200, Cambridge University Press, 2013

    work page 2013

  43. [43]

    Mikhail M Kapranov, Bernd Sturmfels, and Andrei V Zelevinsky, Quotients of toric varieties, Mathematische Annalen 290 (1991), 643--655

    work page 1991

  44. [44]

    2, 535--563

    Li Li, Wonderful compactification of an arrangement of subvarieties, Michigan Mathematical Journal 58 (2009), no. 2, 535--563. 2595553 (2011f:14086)

    work page 2009

  45. [45]

    MR1786500 (2002m:14044)

    Andrey Losev and Yuri Manin, N ew moduli spaces of pointed curves and pencils of flat connections , Michigan Mathematical Journal 48 (2000), 443--472. MR1786500 (2002m:14044)

    work page 2000

  46. [46]

    241--259

    Antonio Laface and Luca Ugaglia, On intrinsic negative curves, The Art of Doing Algebraic Geometry, Springer, 2023, pp. 241--259

    work page 2023

  47. [47]

    thesis, KTH, 2009

    Christian Lundkvist, Moduli spaces of zero-dimensional geometric objects, Ph.D. thesis, KTH, 2009

    work page 2009

  48. [48]

    David McKinnon, Rindra Razafy, Matthew Satriano, and Yuxuan Sun, On curves with high multiplicity on P(a,b,c) for (a,b,c) 4 , New York J. Math. 27 (2021), 1060--1084. 4290553

    work page 2021

  49. [49]

    11, 5015--5023

    Mircea Musta t a, Multiplier ideals of hyperplane arrangements, Transactions of the American Mathematical Society 358 (2006), no. 11, 5015--5023

    work page 2006

  50. [50]

    3, 419 -- 449

    Irena Peeva and Mike Stillman, Toric Hilbert schemes , Duke Mathematical Journal 111 (2002), no. 3, 419 -- 449

    work page 2002

  51. [51]

    21, 17000--17078

    Luca Schaffler and Jenia Tevelev, Compactifications of moduli of points and lines in the projective plane, International Mathematics Research Notices 2022 (2022), no. 21, 17000--17078

    work page 2022

  52. [52]

    1, 1--28

    Hideyasu Sumihiro, Equivariant completion, Journal of Mathematics of Kyoto University 14 (1974), no. 1, 1--28

    work page 1974

  53. [53]

    4, 1087--1104

    Jenia Tevelev, Compactifications of subvarieties of tori, American Journal of Mathematics 129 (2007), no. 4, 1087--1104

    work page 2007