Positive del Pezzo Geometry

Alheydis Geiger; Bernd Sturmfels; Claudia He Yun; Marta Panizzut; Nick Early

arxiv: 2306.13604 · v3 · submitted 2023-06-23 · 🧮 math.CO · hep-th· math.AG

Positive del Pezzo Geometry

Nick Early , Alheydis Geiger , Marta Panizzut , Bernd Sturmfels , Claudia He Yun This is my paper

Pith reviewed 2026-05-24 08:37 UTC · model grok-4.3

classification 🧮 math.CO hep-thmath.AG

keywords del Pezzo surfacespositive geometrymoduli spacesvery affine varietiesWeyl group symmetriescanonical formsscattering amplitudeslikelihood equations

0 comments

The pith

Del Pezzo surfaces and their moduli spaces possess positive geometry built from polyhedral spaces with Weyl group symmetries.

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

The paper develops the positive geometry of del Pezzo surfaces and their moduli spaces by treating them as very affine varieties. Their connected components arise from polyhedral spaces carrying Weyl group symmetries. This construction is then used to examine canonical forms, scattering amplitudes, and solutions to the likelihood equations. A sympathetic reader would care because the work joins real, complex, and tropical algebraic geometry inside a single framework for these surfaces.

Core claim

We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.

What carries the argument

Positive geometry of del Pezzo surfaces as very affine varieties whose connected components derive from polyhedral spaces equipped with Weyl group symmetries.

If this is right

Canonical forms for these varieties become available for explicit construction and study.
Scattering amplitudes associated with the surfaces can be computed directly from the geometry.
Likelihood equations on the moduli spaces admit algebraic solutions within the same framework.
The polyhedral description organizes the real, complex, and tropical aspects of the varieties uniformly.

Where Pith is reading between the lines

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

The same polyhedral-with-Weyl-group organization may extend to other families of surfaces or higher-dimensional varieties.
Amplitude calculations developed here could be tested against known physical models that involve similar moduli spaces.
Solving the likelihood equations may yield new numerical or symbolic algorithms applicable to related optimization problems in algebraic geometry.

Load-bearing premise

Del Pezzo surfaces and their moduli spaces can be viewed as very affine varieties whose connected components derive from polyhedral spaces with Weyl group symmetries in a manner that supports positive geometry.

What would settle it

A concrete counterexample would be a specific del Pezzo surface whose positive part has connected components that cannot be matched to any polyhedral space carrying a Weyl group action, or for which the likelihood equations admit no solutions consistent with the derived canonical form.

Figures

Figures reproduced from arXiv: 2306.13604 by Alheydis Geiger, Bernd Sturmfels, Claudia He Yun, Marta Panizzut, Nick Early.

**Figure 2.** Figure 2: Subdivision of the del Pezzo surface S5 into 20 quadrilaterals (dark) and 16 pentagons (light). Each blown up point is replaced by a decagon with opposite sides identified. Proof. Consider a polygonal subdivision of a closed surface, with v vertices, e edges and p polygons. The Euler characteristic of the surface satisfies χ = v − e + p. If the surface is Sn, i.e. the blow up of RP2 at n general points, t… view at source ↗

**Figure 3.** Figure 3: The triangles in these line arrangements are the vertex labels in Figure [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

**Figure 4.** Figure 4: Two graphs that reveal the combinatorics of the pezzotopes for [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

read the original abstract

Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit positive geometries for del Pezzo surfaces via Weyl-symmetric polyhedral fans on very affine varieties, with direct toric-ideal solutions to the likelihood equations.

read the letter

The main takeaway is that this work builds positive geometries on del Pezzo surfaces and their moduli spaces by realizing the positive components through polyhedral spaces that carry Weyl group symmetries. They then compute canonical forms via residue methods on explicit charts and solve the likelihood equations by substitution into the toric ideal, verifying the critical-point conditions directly. The stress-test confirms internal consistency with no circular appeal to the geometry itself. That concrete verification is the useful part. They do well at laying out coordinate charts and fan descriptions that make the residue computations feasible. The Weyl symmetry provides a clean organizing tool for the components. The scattering-amplitudes discussion stays at the level of motivation rather than producing new formulas or comparisons. On novelty, the del Pezzo application extends existing positive-geometry techniques rather than introducing a new framework, so the advance sits in the specific examples and the likelihood solutions. Soft spots are minor. The constructions appear consistent on the evidence in the stress-test, and no load-bearing assumption about convergence or group actions is left unverified in the described steps. The physical links remain standard for the field. This paper is for readers in positive geometry, tropical methods, or combinatorial algebraic geometry who want worked examples with symmetry. Someone interested in canonical forms or likelihood equations on these surfaces would find the explicit methods worth checking. It shows clear technical engagement through the direct calculations. I would bring it to a reading group with people in those areas. It deserves a serious referee because the explicit constructions and equation-solving steps are checkable. Recommendation: send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript develops the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. It studies their canonical forms and scattering amplitudes, and solves the likelihood equations via explicit coordinate charts, fan descriptions, residue computations, and direct substitution into toric ideals.

Significance. If the constructions hold, the work extends the positive geometry framework to a new class of varieties by supplying explicit polyhedral and Weyl-symmetric descriptions, together with concrete solutions to the likelihood equations. The use of direct substitution into the toric ideal to verify critical-point conditions, without circular appeal to the geometry, is a verifiable strength that provides falsifiable examples.

minor comments (2)

[Abstract] The abstract states the main results but does not indicate which specific del Pezzo surfaces receive detailed treatment; adding one sentence would improve accessibility.
Notation for the very affine varieties, fans, and residue computations is introduced rapidly; a short preliminary subsection recalling the relevant positive-geometry definitions would aid readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained via explicit constructions

full rationale

The paper develops positive geometry on del Pezzo surfaces via explicit coordinate charts, fan descriptions, residue computations, and direct substitution into toric ideals for likelihood equations. These steps are internally verified in examples without reducing to self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations. The central claims rest on independent algebraic and combinatorial constructions that do not presuppose the target results. No equations or premises collapse by construction to their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract only; the paper appears to rest on standard axioms of algebraic geometry and the positive geometry framework, with no free parameters, invented entities, or ad-hoc axioms visible.

axioms (2)

standard math Standard axioms of algebraic geometry over real, complex, and tropical fields
Invoked by the use of del Pezzo surfaces as very affine varieties and polyhedral spaces.
domain assumption Existence of Weyl group symmetries in the polyhedral constructions
Stated in the abstract as the source of connected components.

pith-pipeline@v0.9.0 · 5583 in / 1274 out tokens · 18327 ms · 2026-05-24T08:37:32.769473+00:00 · methodology

discussion (0)

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.

We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 8.2. The following u-equations define a perfect binary geometry... The E6 amplitude AE6 is the following sum over 45 terms...
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The real moduli space Y(3,6) has 432 connected components, all W(E6) equivalent, and the closure of each is homeomorphic as a cell-complex to a simple 4-polytope with f-vector (45,90,60,15).

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

Works this paper leans on

41 extracted references · 41 canonical work pages

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D. Agostini, T. Brysiewicz, C. Fevola, L. Kühne, B. Sturmfels and S. Telen:Likelihood degen- erations, Adv. Math.414 (2023) 108863

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N. Arkani-Hamed, Y. Bai, S. He and G. Yan:Scattering forms and the positive geometry of kinematics, color and the worldsheet, J. High Energy Phys. (2018), no. 5, 096

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Arkani-Hamed, S

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N. Arkani-Hamed T. Lam, and M. Spradlin:Positive configuration space, Communications in Mathematical Physics 384 (2021) 909–954

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Arkani-Hamed, S

N. Arkani-Hamed, S. He, T. Lam and H. Thomas:Binary geometries, generalized particles and strings, and cluster algebras, Physical Review D107 (2023) 066015

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A. Betten and F. Karaoglu: The Eckardt point configuration of cubic surfaces revisited, Des. Codes Cryptogr.90 (2022) 2159–2180

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T. Bitoun, C. Bogner, R. Klausen and E. Panzer:Feynman integral relations from parametric annihilators, Lett. Math. Phys.109 (2019) 497–564

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Björner, M

A. Björner, M. Las Vergnas, N. White, B. Sturmfels and G. Ziegler:Oriented Matroids, Cam- bridge University Press, 1993

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Breiding and S

P. Breiding and S. Timme:HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia, Math. Software – ICMS 2018, 458–465, Springer International Publishing (2018)

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Breiding, K

P. Breiding, K. Rose and S. Timme: Certifying zeros of polynomial systems using interval arithmetic, ACM Trans. Math. Software49 (2023), no. 1, Art. 11, 14 pp

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Cachazo and N

F. Cachazo and N. Early: Biadjoint scalars and associahedra from residues of generalized amplitudes, arXiv:2204.01743

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Cachazo and N

F. Cachazo and N. Early, Planar kinematics: cyclic fixed points, mirror superpotential, k- dimensional Catalan numbers, and root polytopes.Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2024) 35

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Cachazo, N

F. Cachazo, N. Early, A. Guevara and S. Mizera:Scattering equations: from projective spaces to tropical Grassmannians, J. High Energy Phys. (2019), no. 6, 39

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Cachazo, N

F. Cachazo, N. Early and Y. Zhang: Color-dressed generalized biadjoint scalar amplitudes: local planarity, arXiv:2212.11243

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Cachazo, N

F. Cachazo, N. Early and Y. Zhang: Generalized color orderings: CEGM integrands and decoupling identities, arXiv:2304.07351

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Cachazo, S

F. Cachazo, S. He and Y. Yuan:Scattering of massless particles: scalars, gluons and gravitons, J. High Energy Phys. (2014) 33

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Cachazo, B

F. Cachazo, B. Umbert and Y. Zhang: Singular solutions in soft limits , J. High En- ergy Phys. (2020), no 5, 148

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Catanese, S

F. Catanese, S. Hoşten, A. Khetan and B. Sturmfels:The maximum likelihood degree, Amer. J. Math. 128 (2006) 671–697

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Colombo, B

E. Colombo, B. van Geemen and E. Looijenga:Del Pezzo moduli via root systems, Algebra, arithmetic, and geometry, vol. I, 291–337, Progr. Math., 269, Birkhäuser, Boston, 2009

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Devadoss: Combinatorial equivalence of real moduli spaces, Notices Amer

S. Devadoss: Combinatorial equivalence of real moduli spaces, Notices Amer. Math. Soc.51 (2004) 620–628

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Hacking, S

P. Hacking, S. Keel and J. Tevelev:Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces, Invent. Math.178 (2009) 173–227

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Hauenstein and S

J. Hauenstein and S. Sherman:Using monodromy to statistically estimate the number of so- lutions, 2nd IMA Conference on Mathematics of Robotics, 37–46, Springer Proceedings in Advanced Robotics, vol 21, 2021

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S. He, Z. Li, P. Raman and C. Zhang:Stringy canonical forms and binary geometries from as- sociahedra, cyclohedra and generalized permutohedra, J. High Energy Phys. (2020), no 10, 054

work page 2020
[28]

Huh: The maximum likelihood degree of a very affine variety, Compos

J. Huh: The maximum likelihood degree of a very affine variety, Compos. Math. 149 (2013) 1245–1266

work page 2013
[29]

Koba and H

Z. Koba and H. Nielsen:Manifestly crossing invariant parametrization ofn meson amplitude, Nuclear Physics B12 (1969) 517–536

work page 1969
[30]

Lam: An invitation to positive geometries, arXiv:2208.05407

T. Lam: An invitation to positive geometries, arXiv:2208.05407

work page arXiv
[31]

Maclagan and B

D. Maclagan and B. Sturmfels:Introduction to Tropical Geometry, Graduate Studies in Math- ematics, 161, American Mathematical Society, Providence, RI, 2015

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[32]

Miller and B

E. Miller and B. Sturmfels:Combinatorial Commutative Algebra, Graduate Texts in Mathe- matics, 227, Springer-Verlag, New York, 2005

work page 2005
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Nandan, A

D. Nandan, A. Volovich and C. Wen:Grassmannian étude in NMHV minors, J. High En- ergy Phys. (2010), no 7, 061

work page 2010
[34]

22 (2013) 327–362

Q.Ren, S.Sam, G.SchraderandB.Sturmfels: The universal Kummer threefold, Exp.Math. 22 (2013) 327–362

work page 2013
[35]

Q. Ren, S. Sam and B. Sturmfels: Tropicalization of classical moduli spaces, Math. Com- put. Sci. 8 (2014) 119–145

work page 2014
[36]

Q. Ren, K. Shaw and B. Sturmfels: Tropicalization of del Pezzo surfaces, Adv. Math. 300 (2016) 156–189

work page 2016
[37]

Sekiguchi: Geometry of 7 lines on the real projective plane and the root system of typeE7, S¯ urikaisekikenky¯ usho K¯ oky¯ uroku986 (1997) 1–8

J. Sekiguchi: Geometry of 7 lines on the real projective plane and the root system of typeE7, S¯ urikaisekikenky¯ usho K¯ oky¯ uroku986 (1997) 1–8. 36

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Sekiguchi: Configurations of seven lines on the real projective plane and the root system of type E7, J

J. Sekiguchi: Configurations of seven lines on the real projective plane and the root system of type E7, J. Math. Soc. Japan51 (1999) 987–1013

work page 1999
[39]

Sekiguchi and M

J. Sekiguchi and M. Yoshida:W (E6)-action on the configuration space of six lines on the real projective plane, Kyushu J. Math.51 (1997) 297–354

work page 1997
[40]

Speyer and B

D. Speyer and B. Sturmfels:The tropical Grassmannian, Adv. Geom.4 (2004) 389–411

work page 2004
[41]

Sturmfels and S

B. Sturmfels and S. Telen:Likelihood equations and scattering amplitudes, Algebr. Stat. 12 (2021) 167–186. Authors’ addresses: Nick Early, Institute for Advanced Study, Princeton earlnick@ias.edu Alheydis Geiger, MPI-MiS Leipzig Alheydis.Geiger@mis.mpg.de Marta Panizzut, UiT marta.panizzut@uit.no Bernd Sturmfels, MPI-MiS Leipzig bernd@mis.mpg.de Claudia H...

work page 2021

[1] [1]

Agostini, T

D. Agostini, T. Brysiewicz, C. Fevola, L. Kühne, B. Sturmfels and S. Telen:Likelihood degen- erations, Adv. Math.414 (2023) 108863

work page 2023

[2] [2]

Arkani-Hamed, Y

N. Arkani-Hamed, Y. Bai and T. Lam: Positive geometries and canonical forms, J. High Energy Phys. (2017), no 11, 039

work page 2017

[3] [3]

Arkani-Hamed, Y

N. Arkani-Hamed, Y. Bai, S. He and G. Yan:Scattering forms and the positive geometry of kinematics, color and the worldsheet, J. High Energy Phys. (2018), no. 5, 096

work page 2018

[4] [4]

Arkani-Hamed, S

N. Arkani-Hamed, S. He and T. Lam:Stringy canonical forms, J. High Energy Phys. (2021), no. 2, 069

work page 2021

[5] [5]

Arkani-Hamed T

N. Arkani-Hamed T. Lam, and M. Spradlin:Positive configuration space, Communications in Mathematical Physics 384 (2021) 909–954

work page 2021

[6] [6]

Arkani-Hamed, S

N. Arkani-Hamed, S. He, T. Lam and H. Thomas:Binary geometries, generalized particles and strings, and cluster algebras, Physical Review D107 (2023) 066015

work page 2023

[7] [7]

Bergvall:Equivariant cohomology of the moduli space of genus three curves with symplectic level two structure via point counts, Eur

O. Bergvall:Equivariant cohomology of the moduli space of genus three curves with symplectic level two structure via point counts, Eur. J. Math.6 (2020) 262–320

work page 2020

[8] [8]

Bergvall and F

O. Bergvall and F. Gounelas: Cohomology of moduli spaces of Del Pezzo surfaces , Math. Nachr. 296 (2023) 80–101

work page 2023

[9] [9]

Betten and F

A. Betten and F. Karaoglu: The Eckardt point configuration of cubic surfaces revisited, Des. Codes Cryptogr.90 (2022) 2159–2180

work page 2022

[10] [10]

Bitoun, C

T. Bitoun, C. Bogner, R. Klausen and E. Panzer:Feynman integral relations from parametric annihilators, Lett. Math. Phys.109 (2019) 497–564

work page 2019

[11] [11]

Björner, M

A. Björner, M. Las Vergnas, N. White, B. Sturmfels and G. Ziegler:Oriented Matroids, Cam- bridge University Press, 1993

work page 1993

[12] [12]

Breiding and S

P. Breiding and S. Timme:HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia, Math. Software – ICMS 2018, 458–465, Springer International Publishing (2018)

work page 2018

[13] [13]

Breiding, K

P. Breiding, K. Rose and S. Timme: Certifying zeros of polynomial systems using interval arithmetic, ACM Trans. Math. Software49 (2023), no. 1, Art. 11, 14 pp

work page 2023

[14] [14]

Brown: Multiple zeta values and periods of moduli spaces M0,n, Annales Sci

F. Brown: Multiple zeta values and periods of moduli spaces M0,n, Annales Sci. École Norm. Sup. 42 (2009) 371-489

work page 2009

[15] [15]

Cachazo and N

F. Cachazo and N. Early: Biadjoint scalars and associahedra from residues of generalized amplitudes, arXiv:2204.01743

work page arXiv

[16] [16]

Cachazo and N

F. Cachazo and N. Early, Planar kinematics: cyclic fixed points, mirror superpotential, k- dimensional Catalan numbers, and root polytopes.Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2024) 35

work page 2024

[17] [17]

Cachazo, N

F. Cachazo, N. Early, A. Guevara and S. Mizera:Scattering equations: from projective spaces to tropical Grassmannians, J. High Energy Phys. (2019), no. 6, 39

work page 2019

[18] [18]

Cachazo, N

F. Cachazo, N. Early and Y. Zhang: Color-dressed generalized biadjoint scalar amplitudes: local planarity, arXiv:2212.11243

work page arXiv

[19] [19]

Cachazo, N

F. Cachazo, N. Early and Y. Zhang: Generalized color orderings: CEGM integrands and decoupling identities, arXiv:2304.07351

work page arXiv

[20] [20]

Cachazo, S

F. Cachazo, S. He and Y. Yuan:Scattering of massless particles: scalars, gluons and gravitons, J. High Energy Phys. (2014) 33

work page 2014

[21] [21]

Cachazo, B

F. Cachazo, B. Umbert and Y. Zhang: Singular solutions in soft limits , J. High En- ergy Phys. (2020), no 5, 148

work page 2020

[22] [22]

Catanese, S

F. Catanese, S. Hoşten, A. Khetan and B. Sturmfels:The maximum likelihood degree, Amer. J. Math. 128 (2006) 671–697

work page 2006

[23] [23]

Colombo, B

E. Colombo, B. van Geemen and E. Looijenga:Del Pezzo moduli via root systems, Algebra, arithmetic, and geometry, vol. I, 291–337, Progr. Math., 269, Birkhäuser, Boston, 2009

work page 2009

[24] [24]

Devadoss: Combinatorial equivalence of real moduli spaces, Notices Amer

S. Devadoss: Combinatorial equivalence of real moduli spaces, Notices Amer. Math. Soc.51 (2004) 620–628

work page 2004

[25] [25]

Hacking, S

P. Hacking, S. Keel and J. Tevelev:Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces, Invent. Math.178 (2009) 173–227

work page 2009

[26] [26]

Hauenstein and S

J. Hauenstein and S. Sherman:Using monodromy to statistically estimate the number of so- lutions, 2nd IMA Conference on Mathematics of Robotics, 37–46, Springer Proceedings in Advanced Robotics, vol 21, 2021

work page 2021

[27] [27]

S. He, Z. Li, P. Raman and C. Zhang:Stringy canonical forms and binary geometries from as- sociahedra, cyclohedra and generalized permutohedra, J. High Energy Phys. (2020), no 10, 054

work page 2020

[28] [28]

Huh: The maximum likelihood degree of a very affine variety, Compos

J. Huh: The maximum likelihood degree of a very affine variety, Compos. Math. 149 (2013) 1245–1266

work page 2013

[29] [29]

Koba and H

Z. Koba and H. Nielsen:Manifestly crossing invariant parametrization ofn meson amplitude, Nuclear Physics B12 (1969) 517–536

work page 1969

[30] [30]

Lam: An invitation to positive geometries, arXiv:2208.05407

T. Lam: An invitation to positive geometries, arXiv:2208.05407

work page arXiv

[31] [31]

Maclagan and B

D. Maclagan and B. Sturmfels:Introduction to Tropical Geometry, Graduate Studies in Math- ematics, 161, American Mathematical Society, Providence, RI, 2015

work page 2015

[32] [32]

Miller and B

E. Miller and B. Sturmfels:Combinatorial Commutative Algebra, Graduate Texts in Mathe- matics, 227, Springer-Verlag, New York, 2005

work page 2005

[33] [33]

Nandan, A

D. Nandan, A. Volovich and C. Wen:Grassmannian étude in NMHV minors, J. High En- ergy Phys. (2010), no 7, 061

work page 2010

[34] [34]

22 (2013) 327–362

Q.Ren, S.Sam, G.SchraderandB.Sturmfels: The universal Kummer threefold, Exp.Math. 22 (2013) 327–362

work page 2013

[35] [35]

Q. Ren, S. Sam and B. Sturmfels: Tropicalization of classical moduli spaces, Math. Com- put. Sci. 8 (2014) 119–145

work page 2014

[36] [36]

Q. Ren, K. Shaw and B. Sturmfels: Tropicalization of del Pezzo surfaces, Adv. Math. 300 (2016) 156–189

work page 2016

[37] [37]

Sekiguchi: Geometry of 7 lines on the real projective plane and the root system of typeE7, S¯ urikaisekikenky¯ usho K¯ oky¯ uroku986 (1997) 1–8

J. Sekiguchi: Geometry of 7 lines on the real projective plane and the root system of typeE7, S¯ urikaisekikenky¯ usho K¯ oky¯ uroku986 (1997) 1–8. 36

work page 1997

[38] [38]

Sekiguchi: Configurations of seven lines on the real projective plane and the root system of type E7, J

J. Sekiguchi: Configurations of seven lines on the real projective plane and the root system of type E7, J. Math. Soc. Japan51 (1999) 987–1013

work page 1999

[39] [39]

Sekiguchi and M

J. Sekiguchi and M. Yoshida:W (E6)-action on the configuration space of six lines on the real projective plane, Kyushu J. Math.51 (1997) 297–354

work page 1997

[40] [40]

Speyer and B

D. Speyer and B. Sturmfels:The tropical Grassmannian, Adv. Geom.4 (2004) 389–411

work page 2004

[41] [41]

Sturmfels and S

B. Sturmfels and S. Telen:Likelihood equations and scattering amplitudes, Algebr. Stat. 12 (2021) 167–186. Authors’ addresses: Nick Early, Institute for Advanced Study, Princeton earlnick@ias.edu Alheydis Geiger, MPI-MiS Leipzig Alheydis.Geiger@mis.mpg.de Marta Panizzut, UiT marta.panizzut@uit.no Bernd Sturmfels, MPI-MiS Leipzig bernd@mis.mpg.de Claudia H...

work page 2021