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From Regions to Hodge Structures: The Topological Study of semialgebraic Curves Configurations

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Pith's one-line read For curve arrangements, the paper claims a single weighted node count ψ determines the number of regions, the first homology of the incidence graph, and the second Betti and weight-4 Hodge numbers of the complexified complement.

desk verdict The paper's central region-count theorem is false for its own class of allowed curves—a single line segment gives the wrong count—so the advertised semialgebraic framework collapses, even though the line-arrangement OS-defect and the normal-crossing Hodge decomposition are genuine, salvageable results. read the letter →

arxiv 2607.24437 v2 pith:HCQM52DO submitted 2026-07-27 math.AG math.ATmath.KT

Abolfazl Soltanpour This is my paper · ORCID
classification math.AGmath.ATmath.KT MSC 14H5014P1014H2014C3032S2214F40
keywords semialgebraiccurvescurvearrangementsnodecontributionregioncountingOrlik-SolomonalgebramixedHodgestructuredeletion-restrictionintersectionposet
topics The Hodge Conjecture
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's effort is to show that for finite arrangements of connected semialgebraic curves meeting in ordinary multiple points, essentially the whole global picture — how many regions the curves make, the topology of the curve network, the second Betti number of the complexified complement, and the weight-graded Hodge structure — is controlled by one local number, ψ = ½Σ nᵢ(dᵢ−2), a weighted count of intersection nodes. If this is right, the classical pizza-cutting problem, hyperplane-arrangement deletion-restriction, and mixed Hodge theory of plane curve complements all become facets of a single combinatorial quantity. The paper proves exact formulas f=ψ+2 and f=ψ+κ+1 for region counts, a deletion-restriction recurrence ψ(C)=ψ(C′)+v₀, a factorization criterion for a node-based Orlik–Solomon type algebra, and an exact discrepancy formula for line arrangements showing that nodes of multiplicity at least four are the sole source of deviation from cohomology. A sympathetic reader would care because the claim offers a parameter-free bridge from local intersection data to global topology and Hodge theory.

What carries the argument

The load-bearing object is the node contribution ψ, defined by ψ=½Σ nᵢ(dᵢ−2), where nᵢ is the number of intersection nodes at which dᵢ local branches meet. It is a weighted count of singular points, but it appears in the handshaking lemma as e−v=ψ for the one-dimensional incidence complex, which is exactly what converts Euler characteristic into region and homology formulas. The same number is then carried through the deletion-restriction recurrence and into the mixed Hodge structure via the Euler characteristic, so ψ acts as the single combinatorial parameter connecting real region counts, complex cohomology, and Hodge weights.

What would settle it

Draw a single line segment (a bounded open semialgebraic curve, κ=1, ψ=0). The formula f=ψ+κ+1 predicts two regions, but the complement of a segment in R² is connected, so the actual region count is one; this one example kills Theorem 2.5(ii) as stated for the curve class defined in Section 1.

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Extended reading notes

Core claim

At its center, the paper claims that a configuration of curves is governed by the node contribution ψ=½Σ nᵢ(dᵢ−2) (equivalently Σ(k_x−1) over singular points). This one integer reappears in the region count f=ψ+2 (all closed) or f=ψ+κ+1 (κ open curves), in the homology of the incidence complex H₁ ≅ Z^{ψ+κ+1}, in the deletion-restriction recurrence ψ(C)=ψ(C′)+v₀, and in the algebraic and Hodge setting as b₂=ψ for line arrangements and dim Gr^W_4 H²=ψ. The paper also claims that the absence of triple points is sufficient (but not necessary) for the OS-type algebra to factor through cohomology, and that for line arrangements the difference between the simplified model and H² is exactly Σ_{k_x≥4

Load-bearing premise

The load-bearing premise is that every 'open' curve in the configuration is a two-ended unbounded curve that meets a sufficiently large disk in exactly two points; the paper's stated definition also allows bounded intervals and rays, for which the region-count formula and the H₁ corollary are off by one.

Editorial extensions

If this is right

  • The exact region count of any curve configuration is computable from intersection multiplicities alone: f=ψ+2 for closed curves and f=ψ+κ+1 with κ open curves.
  • Maximal configurations are exactly all-transverse double-point arrangements of the form [(n)_4], so maximum region numbers for arbitrary mixed families of convex and concave polygons follow in closed form.
  • The incidence graph of a configuration has first Betti number ψ+κ+1, and its fundamental group is the free group on that many generators.
  • Deleting a curve adds exactly the number v₀ of singular points on it to ψ, giving a recursive computation ψ(C)=ψ(C′)+v₀ that extends the classical deletion-restriction recurrence to curves.
  • For line arrangements, the simplified node-based algebra overcounts cohomology exactly at nodes with four or more lines, by Σ_{k_x≥4} C(k_x−1,2); without such nodes it is an isomorphism, and for normal-crossing arrangements H² is Hodge-Tate exactly when every component has genus zero.

Reading between the lines

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

  • Editorial extension: if the locality principle holds beyond the paper's hypotheses, the same ψ should determine expected region counts for random curve arrangements, since expectation is linear over nodes; this is testable by simulation.
  • Editorial extension: the paper's own Section 1 definition admits bounded intervals and rays as open curves, for which the formula f=ψ+κ+1 fails by one (a single segment has ψ=0, κ=1, predicted two regions, actual one). A corrected statement would count endpoints or restrict to two-ended curves escaping to infinity.
  • Editorial extension: the appearance of k_x=3 as the unique obstruction in both OS factorization and deletion-restriction projection suggests a common combinatorial origin; one could test whether triple points are also the unique obstruction for the motivic recurrence when tangencies are present.
  • Editorial extension: since ψ=Ψ₁−Ψ₀ is a finite difference of binomial node counts and Ψ₂ is universal among linearly local additive invariants, any other additive invariant of ordinary curve configurations is forced to be a multiple of total pairwise intersections; region counts and Betti numbers are therefore not additive in that sense, which sharpens the sense in which ψ is special.
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Editorial analysis

A structured set of objections, weighed in public.

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

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants appear; ψ is a defined incidence sum, not a fitted parameter. The central claim rests on standard semialgebraic finiteness/triangulability, Bézout, Orlik-Solomon, Deligne's MHS spectral sequence, and Varchenko's formula. The paper adds one fragile ad hoc axiom—large-disk two-point intersection for all open curves—that is false for segments and rays, plus an ordinary-multiple-point domain restriction for the algebraic/Hodge results. The OS-type algebra and defect complex are mathematical definitions with constructive content rather than unexplained postulated entities.

assumptions (6)
  • ad hoc to paper Every open semialgebraic curve is cut twice by a sufficiently large circle (giving the 2κ boundary vertices in Δ(C)).
    Used in Theorem 2.5(ii) and Section 4; false for bounded intervals and rays allowed by the semialgebraic definition in §1.
  • domain assumption All nodes are ordinary multiple points, so each k-fold point contributes k_x−1 to ψ and generates a single Koszul relation.
    Definitions 2.3 and 6.4, Theorem 6.23; tacnodes are separately excluded from the normal-crossing result in Definition 7.9.
  • standard math Bézout's theorem bounds distinct complex intersection points by the product of degrees.
    Used in Proposition 5.9 and the Bézout saturation Corollary 5.10.
  • standard math Orlik-Solomon/Brieskorn computes the cohomology of complex line arrangement complements, and Shapiro's theorem gives purity of the mixed Hodge structure.
    Used in Theorem 6.28 and Corollary 7.3 for arbitrary line arrangements.
  • standard math Deligne's spectral sequence for normal-crossing divisors computes the weight-graded cohomology of the complement.
    Used in Theorem 7.10; requires the normal-crossing position hypotheses of Definition 7.9.
  • standard math Varchenko's Euler characteristic formula for plane curve complements.
    Used in Theorem 5.21 and Theorem 7.2; cited to [14].

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Pith. "Pith review of From Regions to Hodge Structures: The Topological Study of semialgebraic Curves Configurations." pith.science (2026). https://pith.science/paper/HCQM52DO

@misc{pith2026260724437,
  author       = {Pith},
  title        = {Pith review of: From Regions to Hodge Structures: The Topological Study of semialgebraic Curves Configurations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCQM52DO}},
  note         = {Machine review of arXiv:2607.24437}
}
abstract

We develop a combinatorial theory for finite arrangements of connected semialgebraic curves with ordinary multiple intersections, governed by a local node contribution $\psi$ that determines global geometric and topological properties. We prove exact region-count formulas, characterize maximal arrangements, and extend the deletion--restriction recurrence to general curve arrangements. In the algebraic setting, we prove that the absence of triple points ($k_x=3$) is a sufficient condition for the OS-type algebra to factor through cohomology; the converse, however, fails already for line arrangements, where the classical Orlik--Solomon relations ensure factorization even in the presence of triple points. For line arrangements we compute the discrepancy between the simplified OS-model and $H^2$, showing it is governed by nodes with $k_x\ge4$ and equals $\sum_{k_x\ge4}\binom{k_x-1}{2}$. The node contribution appears in the mixed Hodge structure via the Euler characteristic; for arrangements in normal crossing position we compute the full weight decomposition of $H^2$ and show it is Hodge--Tate exactly when every component has genus zero, recovering the line-arrangement case as $\dim\operatorname{Gr}^W_4H^2=\psi$. The defect complex for concurrent lines reveals that exactness obstructions require curve-wise incidence data. Finally, we introduce binomial node invariants $\{\Psi_k\}$, prove $\Psi_2$ is the universal linearly locally additive invariant, and show $\psi=\Psi_1-\Psi_0$.

Figures

Figures reproduced from arXiv: 2607.24437 by the authors.

Figure 1
Figure 1. Example of a configuration [(21)4 : (2)6] obtained from a circle, a hyperbola, a line, an ellipse, and a heptagon [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. For the curves C = n R[x,y] ⟨x2y+xy2−x4−y 4⟩ , R[x,y] ⟨x3+x2y−y⟩ o , the configu￾ration is [(1)8 : (2)4]. Theorem 2.5. Let [(n1)d1 : (n2)d2 : · · · : (nk)dk ] be a configuration of semialgebraic curves over the field R; (i) If every curve in the configuration is closed (homeomorphic to a circle), then the configuration partitions the plane R 2 into f = ψ + 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The configuration [(7)4 : (1)8 : (1)12]. Proof. (⇒) Let the configuration [(n1)d1 : · · · : (nk)dk ] be maximal. Assume, for contra￾diction, that there exists a d-fold node with d ≥ 6 (i.e., at least three curves meet at that point). Choose three distinct curves through that node and a small neighbourhood U of the node. Slightly translate one of the three curves inside U so that it no longer passes through the node … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The local maximal configuration two concave quadrilaterals. Lemma 3.4. For two concave pentagons, the local maximal configuration is [(18)4]. The local maximal configurations of other concave polygons or semialgebraic curves can be determined using the general bounds e…
Figure 4
Figure 4. Figure 4: The local maximal configuration two concave quadrilaterals [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: The local maximal configurtion two concave pentagons. Proof. By Lemma 3.2 (two convex n-gons have local maximal configuration [(2n)4]),a con￾figuration is maximal if and only if it is locally maximal for every pair of curves, Thus for m convex n-gons, there are m 2  u…

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Works this paper leans on

54 extracted references · 4 linked inside Pith

  1. [1]

    Steiner,Einige Gesetze ¨ uber die Theilung der Ebene und des Raumes, J

    J. Steiner,Einige Gesetze ¨ uber die Theilung der Ebene und des Raumes, J. Reine Angew. Math., Vol. 1, pp. 349–364, 1826

  2. [2]

    R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete Mathematics, Addison-Wesley, 2nd ed., 1994

    1994

  3. [3]

    D. O. H. Cutler, J. Karlsson, and N. J. A. Sloane,Cutting a Pancake with an Exotic Knife, arXiv:2511.15864 [math.CO], 2025

    arXiv 2025

  4. [4]

    ˇCern´ y, J

    J. ˇCern´ y, J. K´ ara, D. Kr´ al’, P. Podbrdsk´ y, M. Sot´ akov´ a, and R.ˇS´ amal,On the number of intersections of two polygons, Commentationes Mathematicae Universitatis Carolinae, Vol. 44, no. 2, pp. 217–228, 2003

    2003

  5. [5]

    Ackerman, B

    E. Ackerman, B. Keszegh, and G. Rote,An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons, Discrete & Computational Geometry, Vol. 68, pp. 1049–1077, 2022

    2022

  6. [6]

    Orlik and L

    P. Orlik and L. Solomon,Combinatorial Models for Arrangements of Hyperplanes, Inventiones Mathe- maticae, Vol. 59, pp. 167–189, 1980

    1980

  7. [7]

    Zaslavsky,Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Memoirs of the American Mathematical Society, Vol

    T. Zaslavsky,Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Memoirs of the American Mathematical Society, Vol. 1, No. 154, 1975

    1975

  8. [8]

    Dimca,Hyperplane Arrangements: An Introduction, Springer, Cham, Switzerland, 2017

    A. Dimca,Hyperplane Arrangements: An Introduction, Springer, Cham, Switzerland, 2017

    2017

Show all 54 references
  1. [9]

    Saito,Theory of logarithmic differential forms and logarithmic vector fields, Journal of the Faculty of Science, University of Tokyo, Section IA, Vol

    K. Saito,Theory of logarithmic differential forms and logarithmic vector fields, Journal of the Faculty of Science, University of Tokyo, Section IA, Vol. 27, pp. 265–291, 1980

    1980

  2. [10]

    Deligne,Th´ eorie de Hodge: II, Publications Math´ ematiques de l’IH´ES, Vol

    P. Deligne,Th´ eorie de Hodge: II, Publications Math´ ematiques de l’IH´ES, Vol. 40, pp. 5–57, 1971

    1971

  3. [11]

    Deligne,Th´ eorie de Hodge: III, Publications Math´ ematiques de l’IH´ES, Vol

    P. Deligne,Th´ eorie de Hodge: III, Publications Math´ ematiques de l’IH´ES, Vol. 44, pp. 5–77, 1974

    1974

  4. [12]

    Griffiths and J

    P. Griffiths and J. Harris,Principles of Algebraic Geometry, Pure and Applied Mathematics, John Wiley & Sons, New York, 1978

    1978

  5. [13]

    Voisin,Hodge Theory and Complex Algebraic Geometry, Cambridge Studies in Advanced Mathematics, Vol

    C. Voisin,Hodge Theory and Complex Algebraic Geometry, Cambridge Studies in Advanced Mathematics, Vol. 76 & 77, Cambridge University Press, 2002–2003

    2002

  6. [14]

    A. N. Varchenko,The Euler characteristic of the complement of an algebraic curve, Functional Analysis and Its Applications, Vol. 16, No. 2, pp. 134–135, 1982

    1982

  7. [15]

    Andreotti and T

    A. Andreotti and T. Frankel,The Lefschetz theorem on hyperplane sections, Annals of Mathematics, Vol. 69, No. 3, pp. 713–717, 1959

    1959

  8. [16]

    C. A. M. Peters and J. H. M. Steenbrink,Mixed Hodge Structures, Springer Science & Business Media, 2008

    2008

  9. [17]

    Bochnak, M

    J. Bochnak, M. Coste, and M.-F. Roy,Real Algebraic Geometry, Springer Science & Business Media, 2013

    2013

  10. [18]

    Benedetti and J.-J

    R. Benedetti and J.-J. Risler,Real algebraic and semi-algebraic sets, Actualit´ es Math´ ematiques, Her- mann, Paris, 1990. FROM REGIONS TO HODGE STRUCTURES 71

    1990

  11. [19]

    Coste,An introduction to semialgebraic geometry, Notes, 2002

    M. Coste,An introduction to semialgebraic geometry, Notes, 2002. Available at:https://perso. univ-rennes1.fr/michel.coste/polyens/SAG.pdf

    2002

  12. [20]

    L. P. D. van den Dries,Tame Topology and O-minimal Structures, Cambridge University Press, 1998

    1998

  13. [21]

    Fulton,Algebraic Curves: An Introduction to Algebraic Geometry, Addison-Wesley, 1989

    W. Fulton,Algebraic Curves: An Introduction to Algebraic Geometry, Addison-Wesley, 1989

    1989

  14. [22]

    Brieskorn and H

    E. Brieskorn and H. Kn¨ orrer,Plane Algebraic Curves, Birkh¨ auser Verlag, Basel, 1986

    1986

  15. [23]

    S. Basu, R. Pollack, and M.-F. Roy,Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, Vol. 10, Springer-Verlag, 2nd edition, 2006

    2006

  16. [24]

    Orlik and H

    P. Orlik and H. Terao,Arrangements and Hypergeometric Integrals, Mathematical Society of Japan, 2001

    2001

  17. [25]

    Roberts,On the number of regions into which a plane is divided by lines, Proceedings of the London Mathematical Society, Vol

    S. Roberts,On the number of regions into which a plane is divided by lines, Proceedings of the London Mathematical Society, Vol. 19, pp. 53–56, 1887

  18. [26]

    D. T. Lˆ e and C. Weber,Sur la monodromie des courbes alg´ ebriques, Comptes Rendus de l’Acad´ emie des Sciences Paris, S´ erie A, Vol. 282, pp. 23–26, 1976

    1976

  19. [27]

    Persson,Double sextics and singular K-3 surfaces, inAlgebraic Geometry, Proc

    U. Persson,Double sextics and singular K-3 surfaces, inAlgebraic Geometry, Proc. Conf., Sitges (Barcelona)/Spain 1983, Lecture Notes in Mathematics, Vol. 1124, pp. 262–328, Springer, 1985

    1983

  20. [28]

    Randell,The topology of arrangements of algebraic curves, In:Singularities, Part 2 (Arcata, Calif., 1981), Proc

    R. Randell,The topology of arrangements of algebraic curves, In:Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., Vol. 40, pp. 535–544, Amer. Math. Soc., 1983

    1981

  21. [29]

    Yuzvinsky,Orlik-Solomon algebras in algebra and topology, InArrangements and Local Systems, RIMS Kˆ okyˆ uroku, Vol

    S. Yuzvinsky,Orlik-Solomon algebras in algebra and topology, InArrangements and Local Systems, RIMS Kˆ okyˆ uroku, Vol. 1179, pp. 1–15, 2000

    2000

  22. [30]

    Megyesi,Configurations of conics with many tacnodes, Tohoku Mathematical Journal, Vol

    G. Megyesi,Configurations of conics with many tacnodes, Tohoku Mathematical Journal, Vol. 52, No. 4, pp. 555–577, 2000

    2000

  23. [31]

    M. B. Dillencourt, D. M. Mount, and A. Saalfeld,On the maximum number of intersections of two polyhedra in 2 and 3 dimensions, Proc. 5th Canadian Conference on Computational Geometry (CCCG 1993), Waterloo, ON, pp. 49–54, 1993

    1993

  24. [32]

    Levi,Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade, Ber

    F. Levi,Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade, Ber. Math.-Phys. Kl. S¨ achs. Akad. Wiss. Leipzig, Vol. 78, pp. 256–267, 1926

    1926

  25. [33]

    Gr¨ unbaum,Arrangements and Spreads, CBMS Regional Conference Series in Mathematics, No

    B. Gr¨ unbaum,Arrangements and Spreads, CBMS Regional Conference Series in Mathematics, No. 10, American Mathematical Society, Providence, RI, 1972

    1972

  26. [34]

    Felsner and M

    S. Felsner and M. Scheucher,Arrangements of Pseudocircles: Triangles and Drawings, Discrete & Com- putational Geometry, Vol. 65, pp. 261–278, 2021

    2021

  27. [35]

    Dupont,The Orlik-Solomon model for hypersurface arrangements, Annales de l’Institut Fourier, Vol

    C. Dupont,The Orlik-Solomon model for hypersurface arrangements, Annales de l’Institut Fourier, Vol. 65, no. 6, pp. 2507–2545, 2015. doi:10.5802/aif.2994, arXiv:1302.2103

    2015 arXiv

  28. [36]

    A. Libgober,Characteristic varieties of algebraic curves, inApplications of Algebraic Geometry to Cod- ing Theory, Physics and Computation, NATO Science Series II: Mathematics, Physics and Chemistry, Vol. 36, pp. 215–254, Kluwer Academic Publishers, 2001. arXiv:math/9801070

    2001 arXiv

  29. [37]

    V. I. Arnold,On the cohomology ring of the complement of a plane curve, (in Russian), Uspekhi Mat. Nauk, Vol. 24, no. 2(146), pp. 225–226, 1969

    1969

  30. [38]

    Denham,Orlik-Solomon complex and Milnor fibre homology, Topology and its Applications, Vol

    G. Denham,Orlik-Solomon complex and Milnor fibre homology, Topology and its Applications, Vol. 118, pp. 45–63, 2002

    2002

  31. [39]

    Denham and A

    G. Denham and A. Suciu,Multinets and Milnor fibrations, Proc. London Math. Soc., Vol. 95, pp. 527– 562, 2007

    2007

  32. [40]

    Yoshinaga,Milnor fibers of real line arrangements, Journal of Singularities, Vol

    M. Yoshinaga,Milnor fibers of real line arrangements, Journal of Singularities, Vol. 7, pp. 94–108, 2013

    2013

  33. [41]

    Libgober and S

    A. Libgober and S. Yuzvinsky,Cohomology of Orlik-Solomon algebras and local systems, Compositio Mathematica, Vol. 121, pp. 337–361, 2000

    2000

  34. [42]

    M˘ acinic and S

    A. M˘ acinic and S. Papadima,On the monodromy of Milnor fibers of graphic arrangements, Topology and its Applications, Vol. 156, pp. 301–316, 2009

    2009

  35. [43]

    J. I. Cogolludo-Agust ´ ın and D. Matei,Cohomology algebra of plane curves, weak combinatorial type, and formality, Transactions of the American Mathematical Society, Vol. 364, no. 11, pp. 5765–5790, 2012. arXiv:0711.1951

    2012 arXiv

  36. [44]

    Dimca,Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, New York, 1992

    A. Dimca,Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, New York, 1992

    1992

  37. [45]

    Dimca and A

    A. Dimca and A. Libgober,Regular functions transversal at infinity, Tohoku Mathematical Journal, Vol. 58, No. 4, pp. 549–564, 2006

    2006

  38. [46]

    Hattori,Topology ofC n minus a finite number of affine hyperplanes in general position, Journal of the Faculty of Science, University of Tokyo, Section IA Mathematics, Vol

    A. Hattori,Topology ofC n minus a finite number of affine hyperplanes in general position, Journal of the Faculty of Science, University of Tokyo, Section IA Mathematics, Vol. 22, pp. 205–219, 1975. 72 A. SOLTANPOUR

    1975

  39. [47]

    Hamm,Lokale topologische Eigenschaften komplexer R¨ aume, Mathematische Annalen, Vol

    H. Hamm,Lokale topologische Eigenschaften komplexer R¨ aume, Mathematische Annalen, Vol. 191, pp. 235–252, 1971

    1971

  40. [48]

    Randell,The fundamental group of the complement of a union of complex hyperplanes, Inventiones Mathematicae, Vol

    R. Randell,The fundamental group of the complement of a union of complex hyperplanes, Inventiones Mathematicae, Vol. 69, pp. 103–108, 1982

    1982

  41. [49]

    Brieskorn,Sur les groupes de tresses, S´ eminaire Bourbaki, Vol

    E. Brieskorn,Sur les groupes de tresses, S´ eminaire Bourbaki, Vol. 1971/72, Exp. No. 401, Lecture Notes in Mathematics, Vol. 317, Springer, pp. 21–44, 1973

    1971

  42. [50]

    B. Z. Shapiro,The mixed Hodge structure of the complement to an arbitrary arrangement of affine complex hyperplanes is pure, Proceedings of the American Mathematical Society, Vol. 117, No. 4, pp. 931–933, 1993

    1993

  43. [51]

    Milnor,AlgebraicK-theory and quadratic forms, Inventiones Mathematicae, Vol

    J. Milnor,AlgebraicK-theory and quadratic forms, Inventiones Mathematicae, Vol. 9, pp. 318–344, 1970

    1970

  44. [52]

    Weibel,TheK-book: An Introduction to AlgebraicK-theory, Graduate Studies in Mathematics, Vol

    C. Weibel,TheK-book: An Introduction to AlgebraicK-theory, Graduate Studies in Mathematics, Vol. 145, AMS, 2013

    2013

  45. [53]

    Rost,Chow groups with coefficients, Documenta Mathematica, Vol

    M. Rost,Chow groups with coefficients, Documenta Mathematica, Vol. 1, pp. 319–393, 1996

    1996

  46. [54]

    Kerz,The Gersten conjecture for MilnorK-theory, Inventiones Mathematicae, Vol

    M. Kerz,The Gersten conjecture for MilnorK-theory, Inventiones Mathematicae, Vol. 175, pp. 1–33, 2009

    2009

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