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Divisorial Persistence and Asymptotic Homology of Analytic Pairs

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Pith's one-line read An analytic pair carries a canonical persistence module — divisorial asymptotic homology — whose minimum is the real log canonical threshold and whose full structure records how exceptional divisors glue; for normal surface germs its critic

desk verdict Ambitious, cleanly defined persistence invariant for analytic pairs, but the surface rigidity theorem rests on a wrong estimate and Theorem E is not actually proved. read the letter →

arxiv 2607.06717 v4 pith:UAWTOP5C submitted 2026-07-07 math.AG

Nivaldo Grulha This is my paper · ORCID
classification math.AG MSC 32S0532S4514E1514B0555N31
keywords divisorialasymptotichomologyreallogcanonicalthresholdhomologicalspectrumpersistencemoduleanalyticpaircriticalweighteddualgraphbirationalinvariancenormalsurfacesingularity
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

This paper tries to establish that numerical invariants of singularities, especially the real log canonical threshold, are only the first visible layer of a deeper topological structure. It constructs divisorial asymptotic homology (DAH), a persistence module attached canonically to an analytic pair through the sublevel sets of the intrinsic energy. Each chain receives a valuative admissibility threshold, and the resulting module has the RLCT as its minimum and a finite homological spectrum as its jump locus. For normal surface germs, the critical DAH groups are identified with the homology of critical weighted dual graphs, and examples show pairs with identical homological spectra but non-isomorphic DAH. A sympathetic reader should take away that the gluing of exceptional divisors, information discarded by exponents and spectra, is recoverable by a persistence theory whose filtration is determined by birational geometry rather than by a user-chosen scale.

What carries the argument

The central object is the admissibility threshold δ(c), defined purely valuatively as the infimum of divisorial exponents γ(v)=(a(v)+1)/(2ν(v)) over divisorial valuations reaching the chain, equivalently the minimum γ_E over exceptional divisors meeting the total transform on any log-resolution. This threshold filters the chain complex into α-admissible subcomplexes, whose homology defines the persistence module P(X,I). The divisorial exponents themselves, extracted from the order ν_E and Jacobian discrepancy a_E of each exceptional divisor, together with the weighted monomial integral estimates producing ε^{γ}|log ε|^{m−1}, carry the analytic content; in dimension two the critical weighted

What would settle it

For a normal surface germ with a critical divisor E having a_E>0, compute the homotopy-cylinder measure used in Lemma 4.3: H_{k+1}(|r(c)|∩U_ε) ≈ ε^{γ_E}∫_{ε^{1/(2ν_E)}}^1 t^{-(a_E+1)}dt. If for a_E>0 this quantity is not O(ε^{γ_i}), the claimed isomorphism DAH^{(i)}_k ≅ H_k(Γ(i)) fails at that level; a concrete check is to compare the resulting DAH groups for an A_n or cusp pair whose critical divisor has positive discrepancy against the homology of its critical dual graph.

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Extended reading notes

Core claim

The paper claims that for an analytic pair the filtration by sublevel sets of the energy induces a divisorial persistence module whose critical values are exactly the divisorial levels γ_E=(a_E+1)/(2ν_E); the RLCT is its minimum, the homological spectrum is its jump locus, and the whole module is birationally invariant. In dimension two, the critical DAH groups are canonically and resolution-independently isomorphic to the homology of the critical weighted dual graph, making DAH an explicit combinatorial invariant of normal surface germs. The paper further claims that this persistent structure is strictly finer than the RLCT and the homological spectrum, exhibiting pairs with equal homologic

Load-bearing premise

The load-bearing premise is that admissibility survives localization to the critical exceptional set: the radial contraction homotopy in Lemma 4.3 and Proposition 4.4 is claimed to stay γ_i-admissible via the estimate ∫_{t_0}^1 t^{-(a_E+1)}dt ≲ |log ε|, but this bound appears to hold only for a_E=0; for a_E>0 the integral contributes ε^{-a_E/(2ν_E)}, so without an added hypothesis the isomorphism DAH^{(i)}_k ≅ H_k(Exc^{(i)}) is not established. The rigidity theorem also assum

Editorial extensions

If this is right

  • The RLCT becomes the first level of a canonical hierarchy, with min Γ_H(X,I)=1/2 rlct(I).
  • DAH is tame: the persistence module changes only at the finitely many critical values of the homological spectrum.
  • For normal surface germs, the persistence module reduces to the homology of critical weighted dual graphs, making it computable and resolution-independent.
  • DAH is strictly finer than the homological spectrum and the RLCT: pairs with identical spectral data can have non-isomorphic DAH groups.
  • DAH is functorial and admits relative and Mayer–Vietoris long exact sequences, so it behaves like a homology theory rather than a single numerical invariant.

Reading between the lines

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

  • If DAH is as robust as claimed, singular learning theory could be stratified beyond the learning coefficient: the full homological spectrum and the birth times of higher-degree classes (as in the cusp example, where H_1 is born only at the last critical level) suggest testable refinements of asymptotic model selection.
  • The surface rigidity result points toward a higher-dimensional extension in which critical DAH groups are homology groups of a critical dual complex recording intersections of all orders; the paper notes this extension is not developed, making it a concrete open problem.
  • The comparison theorems with metric and vanishing homology are explicitly conditional: the metric-homology identification assumes normal embedding plus an asymptotic expansion, and the vanishing-homology correspondence is stated as a conjecture requiring projective-thinness control and flat-limit stability. A careful reader should treat those comparisons as programmatic, not established.
  • The most fragile step is the localization argument for surface rigidity: the radial contraction bound in Lemma 4.3 and Proposition 4.4 may require a_E=0; if the estimate fails for positive discrepancies, Theorem D would need an additional hypothesis on critical divisors.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces Divisorial Asymptotic Homology (DAH), a persistence module associated to an analytic pair (X,I). Chains are assigned a divisorial admissibility threshold δ via their intersection with exceptional divisors of a log-resolution, and the sublevel filtration of the intrinsic energy K_I yields a persistence module whose minimum is (1/2) rlct(I) and whose jump locus is a finite homological spectrum Γ_H. The main results are: tameness (Theorem 3.32), functoriality (Prop. 3.38), relative and Mayer–Vietoris sequences (Thm. 3.39, 3.40), birational invariance (Thm. 3.47), a surface rigidity theorem identifying critical DAH groups with homology of critical dual graphs (Thm. 4.7 / Theorem D), and an example (Theorem E) claiming two pairs with equal Γ_H but non-isomorphic DAH. The paper situates DAH relative to metric and vanishing homology.

Significance. If the main theorems hold, DAH is an original and potentially useful invariant: it upgrades the RLCT from a single number to a persistence module, proves a surface-level rigidity theorem with a clean combinatorial output, and the paper provides multiple checkable examples. Strengths include a purely valuative definition of δ, a proof of resolution-independence of δ via valuation-theoretic arguments (Lemmas 3.41–3.45, Theorem 3.47), and a clearly stated finiteness/tameness result (Prop. 3.22, Thm. 3.32). The claimed comparison with metric homology (Prop. 5.8) is honest about its conditional hypotheses.

major comments (3)
  1. [Lemma 4.3 / Proposition 4.4] The localization step contains a load-bearing estimate that appears incorrect. Lemma 4.3 and Prop. 4.4 bound the homotopy cylinder of r_t(u,w)=(tu,w) by roughly ε^{γ_E} ∫_{t_0}^1 t^{-(a_E+1)} dt with t_0 = ε^{1/(2ν_E)}, and the text asserts this integral is O(|log ε|). This is only true for a_E = 0. For a_E > 0 the integral is ≍ t_0^{-a_E} = ε^{-a_E/(2ν_E)}, so the cylinder exponent is 1/(2ν_E), not γ_E. The condition 1/(2ν_E) ≥ γ_i is not implied by the hypotheses. Thus the null-homology argument for chains near noncritical divisors in Lemma 4.3 is not established, and the isomorphism DAH^{(i)}_k ≅ H_k(Exc^{(i)}) in Prop. 4.4 does not follow. Since Theorem 4.7 / Theorem D and all examples in §4.5 depend on this result, the surface rigidity theorem is not proven as written.
  2. [Proposition 3.38] Functoriality is asserted with the proof: 'Since f is a morphism of analytic pairs, every divisor meeting g f# c also meets e c. Hence δ(f#c) ≥ δ(c).' For a general ideal morphism (only f*J ⊆ I and the energy condition), this incidence claim is not justified. The strict transform behavior under non-submersive maps is delicate; a divisor meeting the push-forward of c need not lift to a divisor meeting e c. This unproved incidence assertion is load-bearing for Theorem B and for the relative/Mayer–Vietoris sequences.
  3. [Section 4.5.3 and Theorem E] The examples in §4.5 are all computed under the a priori identification DAH ≅ H_*(Γ(α)) of Theorem 4.7, so they do not provide independent evidence for the rigidity theorem. Theorem E is stated as a result about pairs with identical Γ_H but non-isomorphic DAH; in the main text I find a description of the pairs only through the critical weighted dual graphs being non-isomorphic. Given that Prop. 4.4 is the only bridge from DAH to these graphs, the theorem inherits the gap in the localization estimate.
minor comments (4)
  1. [Remark 1.6] The remark correctly notes that interval decomposition is unavailable over GrAb, but the paper still speaks of 'the persistence barcode' in §4.5. Either restrict to field coefficients or explain what 'barcode' means for the group-valued module.
  2. [Proposition 3.25] The proof of resolution-independence via K_I^{(1)} ≍ K_I^{(2)} is terse; the line 'because the exponents in E_ρ are non-negative and bounded above by the discrepancies of ρ' is unclear and should be expanded.
  3. [Section 4.1, displayed normal form] After the displayed K_I ∘ π = ∏_{j=1}^k |u_j|^{2ν_{i_j}}, k ∈ {1,2}, the text says this holds 'up to a nowhere-vanishing analytic unit'; the unit is not tracked in the subsequent estimates. This is cosmetic because the unit is bounded, but should be stated.
  4. [Theorem 3.32] The tameness statement is stated for the module over R_{>0}; the proof shows local constancy of each chain group as a function of α. It would help to state explicitly that the module has finite critical values and is isomorphic to a finite-indexed module.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core derivation is independent of its inputs; self-citations are contextual.

full rationale

The derivation chain is not circular. The persistence module P(X,I) is built from the valuative admissibility threshold δ(c) (Definition 3.2), whose values are shown to lie in the divisorial spectrum Γπ(X,I) via Theorem 3.5/3.47, and the critical levels are then γ_E = (a_E+1)/(2ν_E) by the local monomial estimates of Theorem 2.5. The identification min Γ_H(X,I) = (1/2) rlct(I) is an application of the classical resolution formula rlct(I) = min_E (a_E+1)/ν_E (cited to [28,37]) together with the explicitly stated quadratic energy convention γ_E = λ_E/2; it is not a fitted input renamed as a prediction, and the claimed birational invariance is proved from blow-up invariance of δ (Lemmas 3.42–3.45, Theorem 3.47), not imported from the author's prior work. The self-citations [18] and [19] appear as contextual framing (e.g., "homological complement to the asymptotic birational program initiated in [18]") and are not load-bearing for the main theorems. The Section 5 comparison with vanishing homology is explicitly labelled conditional (Proposition 5.9, Conjecture 5.11, Remark 5.12), so no circularity is concealed there. The technical concern raised about the estimate in Lemma 4.3/Proposition 4.4—whether ∫_{t0}^1 t^{-(a_E+1)} dt is always O(|log ε|) when a_E > 0—is a mathematical correctness risk for Theorem D, not a circularity: it does not reduce the claimed isomorphism to its own input or to a self-citation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central construction depends on standard resolution theorems (Hironaka; Bierstone–Milman), the weak factorization of birational maps, o-minimal geometry/geometric measure theory, and normality plus the SNC standing hypothesis for the rigidity theorem. No free parameters are fitted; the factor 2 in γ_E is a quadratic-energy normalization.

assumptions (6)
  • standard math Log-resolutions exist for analytic pairs (Hironaka) and can be chosen functorially (Bierstone–Milman)
    Assumption 1.13; used to define Γ_π and to compute δ(c) in Theorem 3.5.
  • domain assumption Two log-resolutions admit a common resolution by finite sequences of blow-ups along SNC strata of the marked divisor
    Proposition 3.46; essential for birational invariance of δ (Theorem 3.47).
  • domain assumption Jacobian discrepancies are nonnegative and satisfy the transformation law a_F = Σ a_{E_i} + (k-1) under blow-up of an SNC stratum
    Lemma 3.42 and Table 1; needed for the mediant inequality and resolution independence.
  • domain assumption X normal for birational invariance of Γ_H, the RLCT identification, and the surface rigidity theorem
    Standing convention and Proposition 1.22; normality is explicit in Theorem 4.7.
  • domain assumption Supp(π^{-1}I) ⊆ Exc(π) (e.g. I is m_0-primary) so that all critical divisors are exceptional
    Standing hypothesis of Section 4; needed for the critical dual graph and rigidity isomorphism.
  • standard math O-minimality, Hardt triviality, and the area formula provide uniform fiber bounds and subanalytic triangulation
    Appendix A; used in Theorem 2.5 and Theorem 3.15.
invented entities (2)
  • Divisorial Asymptotic Homology (DAH) independent evidence
    purpose: Topological refinement of numerical divisorial invariants; persistence module with RLCT as min and Γ_H as jump locus
    Predicts specific barcodes for An and cusp surface germs (Theorem 4.7/§4.5) and discriminating examples in Theorem E.
  • Homological spectrum Γ_H(X,I) independent evidence
    purpose: Finite birationally invariant set of critical admissibility thresholds realized by chains
    Defined as the image of δ on chains; its values are checkable on resolutions and it appears as the jump locus of the persistence module.

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Pith. "Pith review of Divisorial Persistence and Asymptotic Homology of Analytic Pairs." pith.science (2026). https://pith.science/paper/UAWTOP5C

@misc{pith2026260706717,
  author       = {Pith},
  title        = {Pith review of: Divisorial Persistence and Asymptotic Homology of Analytic Pairs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UAWTOP5C}},
  note         = {Machine review of arXiv:2607.06717}
}
abstract

Numerical divisorial invariants, such as the log canonical threshold (LCT) and the real log canonical threshold (RLCT), capture the asymptotic geometry of an analytic pair $(X,\mathcal I)$ but discard the topology of the exceptional divisor. We introduce Divisorial Asymptotic Homology (DAH), a valuative persistence theory that recovers this missing topological information. The construction assigns to each chain a divisorial admissibility threshold $\delta$, yielding a persistence module whose minimum recovers the RLCT and whose jump locus defines a finite homological spectrum $\Gamma_{\mathrm H}(X,\mathcal I)$. We prove that DAH is tame, functorial, birationally invariant, and satisfies relative and Mayer--Vietoris exact sequences. For normal analytic surface germs, we identify the critical DAH groups with the homology of critical weighted dual graphs. Examples show that while $\Gamma_{\mathrm H}$ records only the critical divisorial levels, DAH also encodes how the corresponding exceptional divisors are glued together, providing a topological refinement of classical numerical divisorial invariants.

Figures

Figures reproduced from arXiv: 2607.06717 by the authors.

Figure 1
Figure 1. illustrates the relation between the exceptional divisors, the crit￾ical exceptional set, and the associated critical dual graph. Exc(π) = S E E E1 γE1 = 1 4 E2 γE2 = 1 3 E3 γE3 = 1 2 E4 γE4 = 3 4 Exc(i) = E1 ∪ E2 dual graph Γ(π) Γ (i) v1 v2 v3 v4 1 4 1 3 1 2 3 4 Legend critical divisor non-critical divisor critical vertex non-critical vertex Γπ(X, I) =  1 4 , 1 3 , 1 2 , 3 4 [PITH_FULL_IMAGE:figures/full_fig_p041… view at source ↗
Figure 1
Figure 1. Critical exceptional divisors appear in blue on the left, and the associated critical dual [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
Figure 1
Figure 1. Critical exceptional divisors appear in blue on the left, and the associated critical dual [PITH_FULL_IMAGE:figures/full_fig_p043_1.png] view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Geometric Criteria for Morrey Admissibility via the Real Log-Canonical Threshold

    math.AP 2026-07 reject novelty 6.0 of 10

    The paper's claimed exact Morrey admissibility threshold via resolution data is invalid: for f=x^2+y^2 it gives 2/3, while the promised RLCT equality gives 1/2.

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Works this paper leans on

55 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [1]

    V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko,Singularities of differentiable maps. Vol. II, Birkhäuser, Basel, 1988

    1988

  2. [2]

    J. M. Aroca, H. Hironaka, and J. L. Vicente,Complex analytic desingularization, Lecture Notes in Math., vol. 1995, Springer, Berlin, 1977

    1995

  3. [3]

    M. F. Atiyah,Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23(1970), 145–150

    1970

  4. [4]

    Birbrair and J.-P

    L. Birbrair and J.-P. Brasselet,Metric homology, Comm. Pure Appl. Math.53(2000), no. 11, 1434–1447

    2000

  5. [5]

    Birbrair and J.-P

    L. Birbrair and J.-P. Brasselet,Metric homology for isolated conical singularities, Bull. Sci. Math.126(2002), 87–95

    2002

  6. [6]

    Bierstone and P

    E. Bierstone and P. D. Milman,Semianalytic and subanalytic sets, Publ. Math. Inst. Hautes Études Sci.67(1988), 5–42

    1988

  7. [7]

    Bierstone and P

    E. Bierstone and P. D. Milman,Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math.128(1997), 207–302

    1997

  8. [8]

    Bivià-Ausina and T

    C. Bivià-Ausina and T. Fukui,Mixed Łojasiewicz exponents and log canonical thresholds of ideals, J. Pure Appl. Algebra220(2016), no. 1, 223–245

    2016

Show all 55 references
  1. [9]

    Carlsson,Topology and data, Bull

    G. Carlsson,Topology and data, Bull. Amer. Math. Soc. (N.S.)46(2009), no. 2, 255–308

    2009

  2. [10]

    J. B. Conway,A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer, New York, 1990

    1990

  3. [11]

    Coste,An introduction to o-minimal geometry, RAAG Notes, Rennes, 2000

    M. Coste,An introduction to o-minimal geometry, RAAG Notes, Rennes, 2000

    2000

  4. [12]

    van den Dries,Tame topology and o-minimal structures, London Math

    L. van den Dries,Tame topology and o-minimal structures, London Math. Soc. Lecture Note Ser., vol. 248, Cambridge Univ. Press, 1998

    1998

  5. [13]

    de Fernex, L

    T. de Fernex, L. Ein, and M. Mustaţă,Multiplicities and log canonical thresholds, J. Algebraic Geom.13(2004), no. 3, 603–615

    2004

  6. [14]

    Denef,Report on Igusa’s local zeta function, Astérisque206(1992), 359–386

    J. Denef,Report on Igusa’s local zeta function, Astérisque206(1992), 359–386

    1992

  7. [15]

    Federer,Geometric measure theory, Grundlehren Math

    H. Federer,Geometric measure theory, Grundlehren Math. Wiss., vol. 153, Springer, 1969

    1969

  8. [16]

    Goresky and R

    M. Goresky and R. MacPherson,Intersection homology theory, Topology19(1980), no. 2, 135–162

    1980

  9. [17]

    Goresky and R

    M. Goresky and R. MacPherson,Intersection homology II, Invent. Math.72(1983), no. 1, 77–129

    1983

  10. [18]

    N. G. Grulha Jr.,On the divisorial geometry of volume asymptotics of sublevel sets, arXiv:2606.30171 [math.AG] (2026)

    2026 arXiv

  11. [19]

    N. G. Grulha Jr. and T. da Silva,An algebraic–theoretic formulation of divisorial asymptotic homology, preprint, 2026. 68

    2026

  12. [20]

    Hatcher,Algebraic Topology, Cambridge Univ

    A. Hatcher,Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002

    2002

  13. [21]

    R. M. Hardt,Stratification of real analytic mappings and images, Invent. Math.28(1975), 193–208

    1975

  14. [22]

    R. M. Hardt,Semi-algebraic local triviality in semi-algebraic mappings, Amer. J. Math.102 (1980), no. 2, 291–302

    1980

  15. [23]

    Hironaka,Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann

    H. Hironaka,Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. (2)79(1964), 109–326

    1964

  16. [24]

    Hironaka,Subanalytic sets, inNumber Theory, Algebraic Geometry and Commutative Alge- bra, Kinokuniya, Tokyo, 1973, pp

    H. Hironaka,Subanalytic sets, inNumber Theory, Algebraic Geometry and Commutative Alge- bra, Kinokuniya, Tokyo, 1973, pp. 453–493

    1973

  17. [25]

    J. A. Howald,Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc.353(2001), no. 7, 2665–2671

    2001

  18. [26]

    Igusa,An introduction to the theory of local zeta functions, AMS/IP Studies Adv

    J.-I. Igusa,An introduction to the theory of local zeta functions, AMS/IP Studies Adv. Math., vol. 14, Amer. Math. Soc., 2000

    2000

  19. [27]

    S. G. Krantz and H. R. Parks,Geometric Integration Theory, Cornerstones, Birkhäuser Boston, Boston, MA, 2008

    2008

  20. [28]

    Kollár,Singularities of pairs, Proc

    J. Kollár,Singularities of pairs, Proc. Sympos. Pure Math.62, Part 1, Amer. Math. Soc., 1997, pp. 221–287

    1997

  21. [29]

    Kollár,Lectures on Resolution of Singularities, Annals of Math

    J. Kollár,Lectures on Resolution of Singularities, Annals of Math. Studies, vol. 166, Princeton Univ. Press, Princeton, NJ, 2007

    2007

  22. [30]

    Kollár,Singularities of the Minimal Model Program, Cambridge Tracts in Math., vol

    J. Kollár,Singularities of the Minimal Model Program, Cambridge Tracts in Math., vol. 200, Cambridge Univ. Press, 2013

    2013

  23. [31]

    Kollár and S

    J. Kollár and S. Mori,Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math., vol. 134, Cambridge Univ. Press, 1998

    1998

  24. [32]

    Kosta and D

    D. Kosta and D. Windisch,Classification of real hyperplane singularities by real log canonical thresholds, SIAM J. Appl. Algebra Geom.10(2026), no. 2, 238–260. doi:10.1137/25M1747841

    2026 doi

  25. [33]

    Lazarsfeld,Positivity in Algebraic Geometry II, Ergebnisse Math

    R. Lazarsfeld,Positivity in Algebraic Geometry II, Ergebnisse Math. Grenzgeb., vol. 49, Springer, 2004

    2004

  26. [34]

    Lion and J.-P

    J.-M. Lion and J.-P. Rolin,Intégration des fonctions sous-analytiques et volumes des sous- ensembles sous-analytiques, Ann. Inst. Fourier (Grenoble)48(1998), no. 3, 755–767

    1998

  27. [35]

    Mustaţă,Jet schemes of locally complete intersection canonical singularities, Invent

    M. Mustaţă,Jet schemes of locally complete intersection canonical singularities, Invent. Math. 145(2001), no. 3, 397–424

    2001

  28. [36]

    Mustaţă,Singularities of pairs via jet schemes, J

    M. Mustaţă,Singularities of pairs via jet schemes, J. Amer. Math. Soc.15(2002), no. 3, 599–615

    2002

  29. [37]

    Mustaţă,IMPANGA lecture notes on log canonical thresholds, inContributions to Algebraic Geometry, Eur

    M. Mustaţă,IMPANGA lecture notes on log canonical thresholds, inContributions to Algebraic Geometry, Eur. Math. Soc., Zürich, 2012, pp. 407–442

    2012

  30. [38]

    de Fernex and R

    T. de Fernex and R. Docampo,Jacobian discrepancies and rational singularities, J. Eur. Math. Soc. (JEMS)16(2014), no. 1, 165–199. 69

    2014

  31. [39]

    J. F. Nash Jr.,Arc structure of singularities, Duke Math. J.81(1995), no. 1, 31–38 (written in 1968, circulated as a preprint)

    1995

  32. [40]

    Ishii and J

    S. Ishii and J. Kollár,The Nash problem on arc families of singularities, Duke Math. J.120 (2003), no. 3, 601–620

    2003

  33. [41]

    A. J. Reguera,A curve selection lemma in spaces of arcs and the image of the Nash map, Compos. Math.142(2006), no. 1, 119–130

    2006

  34. [42]

    Parusiński,Lipschitz stratification of subanalytic sets, Ann

    A. Parusiński,Lipschitz stratification of subanalytic sets, Ann. Sci. Éc. Norm. Supér. (4)27 (1994), no. 6, 661–696

    1994

  35. [43]

    Rudin,Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw–Hill, New York, 1991

    W. Rudin,Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw–Hill, New York, 1991

    1991

  36. [44]

    Saito,On real log canonical thresholds, preprint, 2007, arXiv:0707.2308

    M. Saito,On real log canonical thresholds, preprint, 2007, arXiv:0707.2308

    2007 arXiv

  37. [45]

    Simon,Lectures on geometric measure theory, Proc

    L. Simon,Lectures on geometric measure theory, Proc. Centre Math. Anal., vol. 3, Austral. Nat. Univ., 1983

    1983

  38. [46]

    Valette,Vanishing homology, Selecta Math

    G. Valette,Vanishing homology, Selecta Math. (N.S.)16(2010), no. 2, 267–296

    2010

  39. [47]

    A. N. Varčenko,Newton polyhedra and estimates of oscillatory integrals, Funkcional. Anal. i Priložen.10(1976), no. 3, 13–38

    1976

  40. [48]

    Watanabe,Algebraic geometry and statistical learning theory, Cambridge Monogr

    S. Watanabe,Algebraic geometry and statistical learning theory, Cambridge Monogr. Appl. Comput. Math., vol. 25, Cambridge Univ. Press, 2009

    2009

  41. [49]

    Watanabe,Recent advances in algebraic geometry and Bayesian statistics, Inf

    S. Watanabe,Recent advances in algebraic geometry and Bayesian statistics, Inf. Geom.7 (Suppl. 1), S187–S209 (2024). doi:10.1007/s41884-022-00083-9

    2024 doi

  42. [50]

    C. A. Weibel,An Introduction to Homological Algebra, Cambridge Stud. Adv. Math., vol. 38, Cambridge Univ. Press, 1994

    1994

  43. [51]

    Zomorodian and G

    A. Zomorodian and G. Carlsson,Computing persistent homology, Discrete Comput. Geom.33 (2005), no. 2, 249–274

    2005

  44. [52]

    Birbrair, W

    L. Birbrair, W. D. Neumann, and A. Pichon,The thick-thin decomposition and the bilipschitz classification of normal surface singularities, Acta Math.212(2014), no. 2, 199–256

    2014

  45. [53]

    Maxim,Intersection homology and Alexander modules of hypersurface complements, Com- ment

    L. Maxim,Intersection homology and Alexander modules of hypersurface complements, Com- ment. Math. Helv.81(2006), no. 1, 123–155

    2006

  46. [54]

    Saito,Mixed Hodge modules, Publ

    M. Saito,Mixed Hodge modules, Publ. Res. Inst. Math. Sci.26(1990), no. 2, 221–333

    1990

  47. [55]

    Schürmann,Topology of Singular Spaces and Constructible Sheaves, Monografie Matematy- czne, vol

    J. Schürmann,Topology of Singular Spaces and Constructible Sheaves, Monografie Matematy- czne, vol. 63, Birkhäuser, Basel, 2003. 70

    2003

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.