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A single explicit operator computes the K-theoretic log double ramification class and also yields a GL_r(Z)-invariant product formula.

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2026-08-02 05:20 UTC pith:LOQIUR2D

load-bearing objection K-theoretic log DR class with a genuinely new stack-level Thom-Porteous formula; the formula and product are likely right, but the colimit logK ring rests on a sketched Artin-fan functoriality that needs a real proof. the 4 major comments →

arxiv 2607.13376 v1 pith:LOQIUR2D submitted 2026-07-15 math.AG

On the K-theoretic logarithmic double ramification class

Kamyar Amini , You-Cheng Chou , Leo Herr , David Holmes , Irit Huq-Kuruvilla , Yuan-Pin Lee This is my paper
classification math.AG MSC 14H1014C1714C3514N3514D2319E08
keywords double ramification cyclelogarithmic geometryalgebraic K-theorycolimit log K-theoryThom–Porteous formuladegeneracy locuscompactified Jacobianmoduli of curves
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 constructs a K-theoretic analogue of the logarithmic double ramification cycle, the virtual class of curves on which a given line bundle becomes fiberwise trivial. It establishes two main results: the class lives in the colimit log K-theory ring and satisfies a product formula with GL_r(Z)-invariance—products for the columns of A equal those for the columns of AM and both equal the higher-rank log DR class—and, at positive genus, the class is explicitly computed by applying an operator eG, a Grothendieck-type polynomial built from exterior powers and determinants, to the derived pushforward of the universal line bundle over a compactified Jacobian. The operator is finite, thanks to a new K-theoretic Thom–Porteous formula for algebraic stacks that uses an explicit finite free resolution rather than divergent infinite series. Together these give both a computational formula for K-theoretic DR classes and the correct ring structure needed to multiply them.

Core claim

At positive genus, the structure sheaf of the zero section e: M_{g,n}→J in a compactified Jacobian equals eG([Rπ'_*F]) in K°(J), where eG(E−F) = [O_Y] − [det E]·(λ_t(F∨)/λ_{t^{-1}}(E))_{t^{r1}}. Consequently, after pulling back along the Abel–Jacobi map, the K-theoretic log double ramification class satisfies [DR^log_{g,n,a}]^{ℓvir} = aj^!_a eG(Rπ'_*F). The proof identifies the zero section with the (r1−1)-degeneracy locus of a map of vector bundles whose ranks differ by 1−g, and applies a finite-resolution K-theoretic Thom–Porteous formula valid for algebraic stacks, bypassing the infinite λ/divided-power expansions that diverge there.

What carries the argument

The central object is the operator eG on the K-theory of a stack, defined by eG(E−F) = [O_Y] − [det E]·(λ_t(F∨)/λ_{t^{-1}}(E))_{t^{r1}}, where λ_t is the exterior power series and the subscript means the coefficient of t^{r1}. This operator converts the class of a vector-bundle map to the pushforward of the structure sheaf of the locus where the map does not have maximal rank. Its stack-valid form comes from an explicit finite resolution (the Eagon–Northcott–Buchsbaum–Rim complex) of that structure sheaf, bypassing the infinite λ/divided-power expansions that truncate on schemes but can diverge on algebraic stacks. The surrounding structure is colimit log K-theory logK°(X) = colim over log a

Load-bearing premise

The product formula holds only if the pullback maps between different log refinements of the moduli space are compatible enough to form a true ring; a single incompatible pair of refinements would destroy the product structure.

What would settle it

Using the explicit eG operator, compute both sides of the product formula for g=1, n=2 with interaction matrix A=[[−1,3],[1,−3]] and M=[[−5,2],[−3,1]] (a worked example in the paper); the two products of log DR classes must coincide in logK°(M_{1,2}). Any difference in a computed class would disprove the product formula.

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

If this is right

  • For g>0, the K-theoretic log double ramification class can be computed by evaluating the finite operator eG on Rπ'_*F, without virtual localization or infinite series; the formula is valid on algebraic stacks and in mixed characteristic.
  • The GL_r(Z)-invariance of products means the class attached to a matrix A is unchanged, after base change, under the action of M; this gives a universal identity in logK°(M_{g,n}) that reduces products of DR classes to a single higher-rank class.
  • Because the classes lie in colimit log K-theory, which is a ring, intersections of log DR classes with one another and with other K-theoretic classes are well-defined—something that fails for limit log K-theory—so this ring is the natural home for double DR intersections and quantum K-theoretic integration.
  • The K-theoretic Thom–Porteous formula extends degeneracy-locus computations from schemes to stacks and is valid in mixed characteristic, making the same finite-resolution tool applicable to other moduli stacks where classes of virtual rank zero are not nilpotent.

Where Pith is reading between the lines

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

  • The finite form of eG suggests that all K-theoretic degeneracy loci on algebraic stacks admit finite alternating-sum formulas, so the apparent need for infinite Grothendieck polynomials is a scheme-specific artifact of nilpotence; analogous finite formulas should hold for rank-drop loci beyond the first.
  • Because the r-th root variant pushes down to the same eG formula, one can conjecture a full K-theoretic Pixton formula—[DR^{log}]·ψ^u = r^{u+1} ϵ_* c_{g+u}(−Rπ_* L^{1/r})—with explicit r-dependence for each u>0, parallel to the Chow precursor; this is testable by the same degeneracy-locus method.
  • The reliance on compactified Jacobians is likely temporary; once the logarithmic Picard stack LogPic has sufficiently developed Brill–Noether theory, the same formula should be provable directly on LogPic, eliminating the quasistable-model and admissibility detour.

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

4 major / 5 minor

Summary. The paper constructs a K-theoretic logarithmic double ramification (log DR) class and studies its formal properties. It defines log perfect obstruction theories for the log DR space and the Abel--Jacobi pullback description, proves their equivalence, introduces a colimit log K-theory ring, and establishes a log product formula together with GL_r(Z)-invariance. It also proves a finite K-theoretic Thom--Porteous formula for degeneracy loci on algebraic stacks, using an Eagon--Northcott resolution, and applies it to compactified Jacobians to express the structure sheaf of the zero section as eG([Rπ'_*F]) and the log DR class as an explicit Gysin pullback of this operator. The main theorems are Theorem 1.8, Theorem 4.19, and Theorem 6.9/Proposition 6.14.

Significance. If correct, the paper is a substantial contribution: it gives a K-theoretic refinement of logarithmic double ramification cycles, a product formula in a genuinely ring-valued log K-theory, and an explicit finite Thom--Porteous formula that remains valid on algebraic stacks where the usual infinite Grothendieck-polynomial series can fail to converge. The degeneracy-locus argument in §5--§6 is independent of the target statement and is not circular; in particular, the Eagon--Northcott resolution is a real methodological strength. However, the framework of colimit log K-theory, and hence Theorems 1.3, 1.4 and 4.19, rests on the Artin-fan functoriality claims in Appendix B, whose proofs are currently not complete. The central formula in Theorem 1.8 also requires clarification of the K-theory group in which the class [Rπ'_*F] is interpreted. For these reasons the paper needs substantial revision before I can recommend acceptance.

major comments (4)
  1. [Appendix B, Proposition B.7; Definition 4.3; Theorem 4.19] The definition of colimit log K-theory and the ring structure on logK^0(X) depend on the existence, uniqueness and compatibility of maps between Artin fans: these are the maps φ: Θ_{X1}→Θ_{X2} used to define the Gysin pullbacks φ^! in Definition 4.3 and in the proof of Theorem 4.4. Proposition B.7 is therefore load-bearing. Its proof is a sketch: Lemma B.6 asserts a uniqueness/extension statement for sheaves on Spec P, but the proof claims that global sections are determined by the stalk at the deepest closed stratum, which is false for general constructible sheaves (e.g. a skyscraper supported on a higher stratum has zero stalk at the deepest stratum but nonzero global sections). The statement of Lemma B.6 also does not assume the constructibility or constancy hypotheses used in the proof. Lemma B.5 similarly asserts an induced étale representable map without a complete verification of
  2. [§6, proof of Theorem 6.9 and Proposition 6.14] The identification of the degeneracy locus with the zero section E^{1/r}→J^{1/r} is imported: the proof says 'which [CH25, Lemma 4.4] then identifies with the zero locus', and Proposition 6.14 uses '[CH25, Lemma 4.5]' to identify a gerbe. These identifications are central to Theorem 1.8. The paper should either state these lemmas with their precise hypotheses and prove them in the required generality, or give a complete reference and explain why the cited statement applies to the present compactified-Jacobian setting. As written, a key step of the main formula is outsourced.
  3. [Definitions 1.5 and Theorem 6.9; Eq. (35)] The operator eG is defined on differences E−F of vector bundles. In Theorem 6.9 it is applied to [Rπ'_*F]=[π'_*F]−[R^1π'_*F]. The proof derives equality of this class with [π'_*F(D)]−[π'_*F_D(D)], where the latter two are vector bundles, but the equality is obtained in G-theory. Unless the cohomology sheaves are locally free, or K^0 is interpreted as K-theory of perfect complexes and eG is extended to that group, the formula eG([Rπ'_*F]) is not literally well-defined as stated. Please clarify which K-group is used throughout §6 and prove that eG extends to the relevant perfect-complex class.
  4. [Theorem 4.19 and §4.3] The proof of the product formula invokes a pullback square involving the simultaneous triviality locus and then applies '[CHL23, Remark 1.8]' to identify ∆^†_B(⊠_i [DR_{P_i,a_i}]^{ℓvir}) with [DR_{P,A}]^{ℓvir}. This is a substantial compatibility statement between the log Gysin map and the log perfect obstruction theories. The paper should justify this square and the obstruction-theory identification in detail, or state the exact theorem from [CHL23] being used.
minor comments (5)
  1. [§1, Theorem 1.8] The notation \(\tilde M_{g,n}\) appears in the statement without definition; it is later described as a log alteration. Please define it in the glossary or in the theorem statement.
  2. [§5, Lemma 5.2] The proof cites '[hg]' (a MathOverflow answer) with only an initial; please give the full author name and a stable reference, or replace by a direct proof.
  3. [§1.6] The paragraph on motivations from quantum K-theory and physics is long and disconnected from the mathematical content. It could be condensed to a few sentences or moved to a remarks subsection.
  4. [§4.3, proof of Theorem 4.19] The statement 'By Remarks 2.18, 2.19 and their analogue in higher rank, we have a pullback square' would be easier to check if the square were written explicitly with arrows and the maps identified.
  5. [§5, Corollary 5.5] The corollary states that it suffices that the grade of the ideal sheaf I_{X/Y} is r_2−r_1+1, but the preceding proof assumes a regular immersion. Please include a short explanation of the grade condition and its compatibility with the determinantal complex.

Circularity Check

0 steps flagged

No significant circularity: the central formula [e]=eG(Rπ'_∗F) is derived from an independent Eagon–Northcott resolution and degeneracy-locus identification, not from the target class.

full rationale

The paper's main derivation is not a restatement of its inputs. Theorem 1.8 is obtained by writing the universal line bundle on a compactified Jacobian, forming the exact sequence (34)–(35), observing via Lemma 6.11 that the ranks satisfy r1−r2 = 1−g, and then applying the finite Eagon–Northcott/Buchsbaum–Rim resolution (Theorem 5.1, Corollary 5.5) to identify the K-theory class of the zero section. The identification of the degeneracy locus with the zero locus is an external geometric statement cited from [CH25], not from the class being computed. The log product formula (Theorem 4.19) is derived from the pullback square and the log Gysin map, with compatibility facts cited from [CHL23]/[Her23]; these are published prior theorems with stated assumptions, so under the rules they count as independent support rather than circularity. The appendix's Artin-fan functoriality lemmas (B.5–B.6) are the weakest point and were revised after an AI-assisted error correction, but even if those proofs are incomplete, the failure would be a gap in the construction of the ring, not a reduction of the conclusion to its inputs. No parameter is fitted and no 'prediction' is statistically forced by construction. Hence no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No numerical free parameters. The paper introduces no new physical entities; the operator eG is a notation, not an entity. All axioms are imported from (or stated variants of) established log-geometry theorems.

axioms (5)
  • domain assumption Log perfect obstruction theory compatibility under log alterations (Theorem B.1, [Her23 Thm 3.10], [CHL23 Prop 2.5])
    Imported theorem underpinning the definition and pushforward-compatibility of the log virtual classes in §4.1; without it, [Q]^{ℓvir} is not well-defined as a compatible system.
  • domain assumption Bounded functoriality of Artin fans: subdivision theorem [ACWM17 Thm 4.6.2] and existence of Artin-fan maps Θ_X→Θ_Y (Prop B.7)
    Needed to define the colimit logK ring and Gysin pullbacks between log alterations; proof in Appendix B relies on Lemmas B.5-B.6, which the acknowledgments say were corrected after an AI-assisted check.
  • domain assumption Log Picard stack properties from [MW22]: exact sequence 1→Pic[0]→LogPic→TroPic→0, representability of diagonal of LP^{ps}_{g,n}
    Used in §2 to define the Abel-Jacobi map and the DR fiber product; not re-proved.
  • domain assumption Existence of compactified Jacobians and universal admissible line bundle (Definition 2.24, [HKP18], [HMP+25])
    The explicit formula in §6 proceeds via J rather than LogPic, because the paper notes LogPic lacks developed Brill-Noether theory (§1.5, §6).
  • domain assumption Identification of the r-th root zero locus with the degeneracy locus of π'_*F^{1/r}(D) (Lemma 4.4 of [CH25])
    Makes the square (33) cartesian; if this identification fails outside Pic^[0], Theorem 6.9 would not compute [e^{1/r}].

pith-pipeline@v1.3.0-alltime-deepseek · 40770 in / 18392 out tokens · 174871 ms · 2026-08-02T05:20:49.183345+00:00 · methodology

0 comments
read the original abstract

The logarithmic double ramification cycle is the virtual fundamental class of the locus where a line bundle on a family of curves is fiberwise trivial. We construct a K-theoretic logarithmic double ramification class and prove a product formula and a \(\mathrm{GL}_r(\mathbb Z)\)-invariance property. We also give an explicit formula for this class in terms of a Grothendieck polynomial via a novel $K$-theoretic Thom--Porteous formula for vector bundles on algebraic stacks.

Figures

Figures reproduced from arXiv: 2607.13376 by David Holmes, Irit Huq-Kuruvilla, Kamyar Amini, Leo Herr, You-Cheng Chou, Yuan-Pin Lee.

Figure 1
Figure 1. Figure 1: The blowup Be → B = M1,2 through which DRlog OC (2p−2q) factors as a strict closed immersion. e1 t e2 e e1 1 t e2 e1 t e2 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Artin fans associated to the log blowup Be → B. The Artin fan of B consists of two cones joined along a ray, depicted in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The tropical abelian variety from Example 2.13 is the “blow down” in log geometry of a nodal cubic or the curve C ′ 0 from [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A family of divisors Ds on a curve C/S. These smooth points Ds ∈ Cs correspond to a line bundle Ls on Cs. Taking D0 ∈ C0 to be the limit of the points Ds, we see it is not a smooth point and does not correspond to a line bundle on C0. Blow up the point D0 ∈ C to obtain a quasistable model C ′ := BlD0C → C which admits a limiting divisor D′ 0 ∈ C ′ 0 of the family Ds. Remark 2.23. The moduli problem Picqs c… view at source ↗
Figure 5
Figure 5. Figure 5: A commutative, noncartesian diagram of the zero sections of J and J 1/r . Stability and marked points are irrelevant, so it suffices to show the same claim for prestable curves Mg,0(r) → Mg,0 with no marked points. Etale localize to replace ´ Mg,0 by S := A k , with coordinates t1, · · · , tk corresponding to smoothing parameters of the k nodes of a vertical log curve C/S. There is exactly one root stack C… view at source ↗

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Reference graph

Works this paper leans on

142 extracted references · 18 canonical work pages · 2 internal anchors

  1. [1]

    Anderson, David , TITLE =. Adv. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.aim.2019.04.049 , URL =

    work page doi:10.1016/j.aim.2019.04.049 2019

  2. [2]

    Using Stacks to Impose Tangency Conditions on Curves , urldate =

    Charles Cadman , journal =. Using Stacks to Impose Tangency Conditions on Curves , urldate =

  3. [3]

    Michigan Mathematical Journal , number =

    Alexander Givental , title =. Michigan Mathematical Journal , number =. 2000 , doi =

    2000

  4. [4]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Obstruction theories and virtual fundamental classes. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  5. [5]

    1995 , isbn =

    David Eisenbud , title =. 1995 , isbn =

    1995

  6. [6]

    2003 , publisher =

    Cohomology of Vector Bundles and Syzygies , author =. 2003 , publisher =

    2003

  7. [7]

    Mathematische Annalen , year=

    Deformation theory of singular symplectic n-folds , author=. Mathematische Annalen , year=

  8. [8]

    Mixed Hodge polynomials of character varieties , volume =

    Hausel, Tamas and Rodriguez-Villegas, Fernando , year =. Mixed Hodge polynomials of character varieties , volume =. Inventiones mathematicae , note =

  9. [9]

    Osaka Journal of Mathematics , year=

    The Riemann-Roch theorem for complex V -manifolds , author=. Osaka Journal of Mathematics , year=

  10. [10]

    https://mathoverflow.net/q/432310 , URL =

    Is the monoid of all cancellative finitely generated commutative monoids cancellative? , AUTHOR =. https://mathoverflow.net/q/432310 , URL =

  11. [11]

    Refined Tropicalizations for Sch\"on Subvarieties of Tori

    Refined Tropicalizations for Sch \"o n Subvarieties of Tori. arXiv e-prints , keywords =. doi:10.48550/arXiv.1705.05719 , archivePrefix =. 1705.05719 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1705.05719

  12. [12]

    Algebra & Number Theory , keywords =

    The log product formula. Algebra & Number Theory , keywords =

  13. [13]

    Hudson, Thomas and Ikeda, Takeshi and Matsumura, Tomoo and Naruse, Hiroshi , TITLE =. Adv. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.aim.2017.08.038 , URL =

    work page doi:10.1016/j.aim.2017.08.038 2017

  14. [14]

    Mixed Hodge structures associated with isolated singularities , volume =

    Steenbrink, Joseph , year =. Mixed Hodge structures associated with isolated singularities , volume =. Proc. Symp. Pure Math. , doi =

  15. [15]

    Bulletin de la Soci\'et\'e Math\'ematique de France , pages =

    Du Bois, Philippe , title =. Bulletin de la Soci\'et\'e Math\'ematique de France , pages =. 1981 , doi =

    1981

  16. [16]

    2014 , publisher=

    Introduction to Singularities , author=. 2014 , publisher=

    2014

  17. [17]

    Homological Bondal-Orlov localization conjecture for rational singularities , volume =

    Mauri, Mirko and Shinder, Evgeny , year =. Homological Bondal-Orlov localization conjecture for rational singularities , volume =. Forum of Mathematics, Sigma , doi =

  18. [18]

    arXiv e-prints , keywords =

    Remarks on logarithmic \'e tale sheafification. arXiv e-prints , keywords =. doi:10.48550/arXiv.2311.05172 , archivePrefix =. 2311.05172 , primaryClass =

    Pith/arXiv arXiv doi:10.48550/arxiv.2311.05172

  19. [19]

    Documenta Mathematica , volume =

    Anderson, Dave and Payne, Sam , title =. Documenta Mathematica , volume =. 2015 , pages =

    2015

  20. [20]

    Journal f\"ur die reine und angewandte Mathematik , volume =

    Brion, Michel and Vergne, Mich\`ele , title =. Journal f\"ur die reine und angewandte Mathematik , volume =. 1997 , pages =

    1997

  21. [21]

    arXiv e-prints , keywords =

    Log geometry and lifting rational points. arXiv e-prints , keywords =. doi:10.48550/arXiv.2509.12167 , archivePrefix =. 2509.12167 , primaryClass =

    work page doi:10.48550/arxiv.2509.12167

  22. [22]

    Compositio Mathematica , author=

    The logarithmic Picard group and its tropicalization , volume=. Compositio Mathematica , author=. 2022 , pages=. doi:10.1112/S0010437X22007527 , number=

    work page doi:10.1112/s0010437x22007527 2022

  23. [23]

    Intersection Theory of Moduli Space of Stable N-Pointed Curves of Genus Zero , volume =

    Keel, Sean , doi =. Intersection Theory of Moduli Space of Stable N-Pointed Curves of Genus Zero , volume =. Transactions of the American Mathematical Society , keywords =

  24. [24]

    Skeletons and Fans of Logarithmic Structures , isbn =

    Abramovich, Dan and Chen, Qile and Marcus, Steffen and Ulirsch, Martin and Wise, Jonathan , year =. Skeletons and Fans of Logarithmic Structures , isbn =

  25. [25]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    The Evaluation Space of Logarithmic Stable Maps. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  26. [26]

    1989 , journal =

    Kazuya Kato , title =. 1989 , journal =

    1989

  27. [27]

    and Moret-Bailly, L

    Laumon, G. and Moret-Bailly, L. , isbn=. Champs alg. 1999 , publisher=

    1999

  28. [28]

    Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds

    Li, An-Min and Ruan, Yongbin. Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds. Inventiones mathematicae. 2001. doi:10.1007/s002220100146

    work page doi:10.1007/s002220100146 2001

  29. [29]

    Communications in Mathematical Physics , volume =

    Kontsevich, Maxim and Manin, Yuri , title =. Communications in Mathematical Physics , volume =. 1994 , pages =

    1994

  30. [30]

    Duke Mathematical Journal , volume =

    Behrend, Kai and Manin, Yuri , title =. Duke Mathematical Journal , volume =. 1996 , pages =

    1996

  31. [31]

    Logarithmic

    Márton Hablicsek and Leo Herr and Francesca Leonardi , keywords =. Logarithmic. Journal of Algebra , volume =. 2026 , issn =. doi:https://doi.org/10.1016/j.jalgebra.2025.07.048 , url =

    doi:10.1016/j.jalgebra.2025.07.048 2026

  32. [32]

    Parker , journal =

    Eleny-Nicoleta Ionel and Thomas H. Parker , journal =. Relative Gromov-Witten Invariants , volume =

  33. [33]

    Parker , journal =

    Eleny-Nicoleta Ionel and Thomas H. Parker , journal =. The Symplectic Sum Formula for Gromov-Witten Invariants , volume =

  34. [34]

    Stable Morphisms to Singular Schemes and Relative Stable Morphisms

    Li, Jun. Stable Morphisms to Singular Schemes and Relative Stable Morphisms. J. Differential Geom. 2001. doi:10.4310/jdg/1090348132

    arXiv 2001

  35. [35]

    A Degeneration Formula of GW-Invariants

    Li, Jun. A Degeneration Formula of GW-Invariants. J. Differential Geom. 2002. doi:10.4310/jdg/1090351102

    arXiv 2002

  36. [36]

    On Log Flat Descent

    Illusie, Luc and Nakayama, Chikara and Tsuji, Takeshi , journal =. On Log Flat Descent. 2013 , volume =

    2013

  37. [37]

    G\'eometrie Logarithmique

    Luc Illusie. G\'eometrie Logarithmique. Notes of a course at IHP. 1997

    1997

  38. [38]

    Logarithmic Geometry and Algebraic Stacks

    Martin Olsson. Logarithmic Geometry and Algebraic Stacks. Scientific Annals of the Ecole Normale Sup\'erieure. 2003

    2003

  39. [39]

    The Logarithmic Cotangent Complex

    Martin Olsson. The Logarithmic Cotangent Complex. Mathematische Annalen. 2005

    2005

  40. [40]

    Sheaves on Artin stacks , volume =

    Olsson, Martin , year =. Sheaves on Artin stacks , volume =. Journal Fur Die Reine Und Angewandte Mathematik - J REINE ANGEW MATH , doi =

  41. [41]

    arXiv e-prints , keywords =

    Logarithmic compactification of the A bel- J acobi section. arXiv e-prints , keywords =

  42. [42]

    Annales de l'Institut Fourier , publisher =

    Abramovich, Dan and Marcus, Steffen and Wise, Jonathan , title =. Annales de l'Institut Fourier , publisher =. 2014 , pages =. doi:10.5802/aif.2892 , mrnumber =

    doi:10.5802/aif.2892 2014

  43. [43]

    Birational invariance in logarithmic

    Abramovich, Dan and Wise, Jonathan , year=. Birational invariance in logarithmic. Compositio Mathematica , publisher=. doi:10.1112/S0010437X17007667 , number=

    doi:10.1112/s0010437x17007667

  44. [44]

    Groupes de monodromie en g\'eom\'etrie alg\'ebrique

    Grothendieck, Alexander and Raynaud, Michel and Rim, Dock Sang. Groupes de monodromie en g\'eom\'etrie alg\'ebrique. I. 1972

    1972

  45. [45]

    Inventiones Mathematicae , keywords =

    Gromov-Witten invariants in algebraic geometry. Inventiones Mathematicae , keywords =. 1997. doi:10.1007/s002220050132 , archivePrefix =. alg-geom/9601011 , primaryClass =

    Pith/arXiv arXiv 1997

  46. [46]

    Inventiones Mathematicae , volume =

    Behrend, Kai and Fantechi, Barbara , title =. Inventiones Mathematicae , volume =. 1997 , doi =

    1997

  47. [47]

    Lectures on Logarithmic Algebraic Geometry , DOI=

    Ogus, Arthur , year=. Lectures on Logarithmic Algebraic Geometry , DOI=

  48. [48]

    Tannaka duality revisited , journal =

    Bhargav Bhatt and Daniel Halpern-Leistner , keywords =. Tannaka duality revisited , journal =. 2017 , issn =. doi:https://doi.org/10.1016/j.aim.2016.08.040 , url =

    doi:10.1016/j.aim.2016.08.040 2017

  49. [49]

    Duke Mathematical Journal , number =

    Tommaso de Fernex and Roi Docampo , title =. Duke Mathematical Journal , number =. 2020 , doi =

    2020

  50. [50]

    and Qu, F

    Lee, Y.-P. and Qu, F. A Product Formula for Log Gromov-Witten Invariants. J. Math. Soc. Japan. 2018. doi:10.2969/jmsj/07017521

    arXiv 2018

  51. [51]

    Algebra & Number Theory , number =

    Jack Hall and David Rydh , title =. Algebra & Number Theory , number =. 2019 , doi =

    2019

  52. [52]

    2005 , publisher=

    Fundamental Algebraic Geometry: Grothendieck's FGA Explained , author=. 2005 , publisher=

    2005

  53. [53]

    arXiv e-prints , keywords =

    The Scheme of Monogenic Generators I: Representability. arXiv e-prints , keywords =

  54. [54]

    eprint arXiv:alg-geom/9710014 , journal =

    The product formula for Gromov-Witten invariants. eprint arXiv:alg-geom/9710014 , journal =. 1999 , volume =

    Pith/arXiv arXiv 1999

  55. [55]

    Moduli of morphisms of logarithmic schemes , volume =

    Wise, Jonathan , year =. Moduli of morphisms of logarithmic schemes , volume =. Algebra & Number Theory , doi =

  56. [56]

    Clutching and gluing in tropical and logarithmic geometry , volume =

    Huszar, Alana and Marcus, Steffen and Ulirsch, Martin , year =. Clutching and gluing in tropical and logarithmic geometry , volume =. Journal of Pure and Applied Algebra , doi =

  57. [57]

    Eagon, J. A. and Northcott, Douglas Geoffrey , title =. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences , volume =. 1962 , month =. doi:10.1098/rspa.1962.0170 , url =

    arXiv 1962

  58. [58]

    Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford , volume =

    Toen, Bertrand , year =. Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford , volume =. K-Theory, v.18, 33-76 (1999) , doi =

    1999

  59. [59]

    arXiv e-prints , keywords =

    A moduli stack of tropical curves. arXiv e-prints , keywords =. 2017

    2017

  60. [60]

    1988 , publisher=

    Determinantal Rings , author=. 1988 , publisher=

    1988

  61. [61]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Virtual pull-backs. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  62. [62]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Logarithmic Gromov-Witten invariants. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  63. [63]

    Jockers, Hans and Mayr, Peter , journal =. Quantum. 2019 , number =

    2019

  64. [64]

    A 3d gauge theory/quantum

    Jockers, Hans and Mayr, Peter , journal =. A 3d gauge theory/quantum. 2020 , volume =

    2020

  65. [65]

    Duke Mathematical Journal , number =

    Dan Edidin and William Graham , title =. Duke Mathematical Journal , number =. 2000 , doi =

    2000

  66. [66]

    Compositio Mathematica , year =

    Roy Joshua , title =. Compositio Mathematica , year =. doi:10.1023/A:1022849624526 , url =

    work page doi:10.1023/a:1022849624526

  67. [67]

    Communications in Mathematical Physics , keywords =

    Double Ramification Cycles and Integrable Hierarchies. Communications in Mathematical Physics , keywords =. doi:10.1007/s00220-014-2235-2 , archivePrefix =. 1403.1719 , primaryClass =

    Pith/arXiv arXiv

  68. [68]

    Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee

    Bumsig Kim and Andrew Kresch and Tony Pantev. Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee. Journal of Pure and Applied Algebra. 2003. doi:https://doi.org/10.1016/S0022-4049(02)00293-1

    work page doi:10.1016/s0022-4049(02)00293-1 2003

  69. [69]

    Johannes Nicaise and Andreas Gross and John Welliaveetil , title =

  70. [70]

    Cycle groups for A rtin stacks

    Kresch, Andrew. Cycle groups for A rtin stacks. Invent. Math. 1999

    1999

  71. [71]

    Log smooth deformation theory

    Kato, Fumiharu. Log smooth deformation theory. Tohoku Math. J. (2). 1996. doi:10.2748/tmj/1178225336

    arXiv 1996

  72. [72]

    Log Smooth Deformation and Moduli of Log Smooth Curves , volume =

    Kato, Fumiharu , year =. Log Smooth Deformation and Moduli of Log Smooth Curves , volume =

  73. [73]

    Costello , year =

    K. Costello , year =. Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products , pages =. Ann. of Math

  74. [74]

    Wies. J. Algebraic Geom. , title =. 2006 , volume =

    2006

  75. [75]

    Journal of the Institute of Mathematics of Jussieu , author=

    EXTENDING THE DOUBLE RAMIFICATION CYCLE BY RESOLVING THE ABEL-JACOBI MAP , volume=. Journal of the Institute of Mathematics of Jussieu , author=. 2021 , pages=. doi:10.1017/S1474748019000252 , number=

    work page doi:10.1017/s1474748019000252 2021

  76. [76]

    https://mathoverflow.net/q/34512 , URL =

    Symmetric powers and duals of vector bundles in char p , AUTHOR =. https://mathoverflow.net/q/34512 , URL =

  77. [77]

    Kazuya Kato , title =

  78. [78]

    Multiplicativity of the Double Ramification Cycle , volume =

    Holmes, David and Pixton, Aaron and Schmitt, Johannes , year =. Multiplicativity of the Double Ramification Cycle , volume =. Documenta Mathematica , doi =

  79. [79]

    2008 , keywords =

    Geometric motivic integration on Artin n-stacks , author =. 2008 , keywords =

    2008

  80. [80]

    2019 , howpublished = "

    Mallory, Devlin , title =. 2019 , howpublished = "

    2019

Showing first 80 references.