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arxiv: 2604.14536 · v1 · submitted 2026-04-16 · 🧮 math.AG · math.AT

Oriented Cohomology Rings of Some Moduli Spaces via Blowups

Arkamouli Debnath , Michael Ruofan Zeng This is my paper

Pith reviewed 2026-05-10 10:09 UTC · model grok-4.3

classification 🧮 math.AG math.AT
keywords oriented cohomologyblowup formulamoduli spacesMbar 0 nChow ringenumerative geometrydel Pezzo surfaces
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The pith

Oriented cohomology theories satisfy an additive blowup formula without A1-invariance, yielding explicit ring presentations for blowups and the moduli space of stable curves.

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

The paper proves that oriented cohomology theories, which include the Chow ring as a special case but extend to non-A1-invariant examples such as topological Hochschild homology, admit an additive formula describing the cohomology of a blowup along a smooth center. This formula is established first in the general non-A1-invariant setting and then specialized to produce concrete ring presentations for blowups of smooth schemes. The same method supplies a presentation for the oriented cohomology ring of the moduli space of stable n-pointed rational curves that reduces to Keel's well-known description when the theory is the Chow ring. A reader should care because the result makes intersection-theoretic calculations available uniformly across a wider collection of theories and recovers classical enumerative counts such as the number of conics tangent to five given conics.

Core claim

We prove an additive blowup formula for oriented cohomology theories in the non-A1-invariant category of motivic spectra. Specializing to A1-invariant theories, we obtain presentations of the oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples for del Pezzo surfaces, the blowup of projective 3-space along the twisted cubic, and the blowup of projective 5-space along the Veronese surface. These presentations allow recovery of classical enumerative numbers such as Steiner's 3264 conics using arbitrary oriented cohomology theories. Finally, we give a presentation of the oriented cohomology rings of the moduli space of stable n-pointed 0

What carries the argument

The additive blowup formula for oriented cohomology theories in the non-A1-invariant motivic spectra category, which describes how the cohomology ring of the blowup decomposes additively in terms of the base and the center.

If this is right

  • Explicit ring presentations become available for the oriented cohomology of del Pezzo surfaces, the blowup of P^3 along the twisted cubic, and the blowup of P^5 along the Veronese surface.
  • Classical enumerative problems such as counting conics tangent to five given conics admit solutions inside any oriented cohomology theory.
  • The Chow ring presentation of the moduli space of stable n-pointed rational curves extends to a presentation valid for every oriented cohomology theory.

Where Pith is reading between the lines

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

  • The blowup formula may supply presentations for oriented cohomology rings of other blowup constructions arising in moduli problems.
  • Uniform computations across different cohomology theories could reveal which enumerative invariants depend only on the underlying geometry rather than the choice of theory.
  • Applications to non-A1-invariant theories might connect classical algebraic geometry counts to invariants appearing in topological cyclic homology.

Load-bearing premise

Oriented cohomology theories in the non-A1-invariant motivic spectra category must exist and satisfy the expected functoriality and excision properties, and both the ambient scheme and the blowup center must be smooth.

What would settle it

An explicit computation of the oriented cohomology ring of the blowup of projective 3-space along the twisted cubic that fails to match the ring structure predicted by the blowup formula.

Figures

Figures reproduced from arXiv: 2604.14536 by Arkamouli Debnath, Michael Ruofan Zeng.

Figure 1
Figure 1. Figure 1: A schematic diagram of the 4 secant lines (red) meeting two general lines (blue) in P3 . We outline the solution using classical Grassmannian Schubert calculus. The secant variety SectpCq – P2 naturally embeds into the Grassmannian Gp1, 3q :“ Grp2, 4q of lines in P3 . The image of Gp1, 3q under the Plücker embedding p is the Plücker quadric in P5 . Thus, we have the composite ι : SectpCq Ñ Gp1, 3q p ÝÑ P 5… view at source ↗
read the original abstract

Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb A^1$-invariance. Topological Hochschild homology, topological cyclic homology, and Hodge cohomology are important examples of theories without $\mathbb A^1$-invariance. In this paper, we prove an additive blowup formula for oriented cohomology theories in the non-$\mathbb A^1$-invariant category of motivic spectra, developed by Annala, Hoyois, and Iwasa. Then, we specialize to $\mathbb A^1$-invariant theories and give presentations of oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces, the blowup of $\mathbb P^3$ along the twisted cubic, and the blowup of $\mathbb P^5$ along the Veronese surface, which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to classical enumerative geometry problems, such as Steiner's $3264$ conics, using arbitrary oriented cohomology theories. Finally, we give a presentation of oriented cohomology rings of $\overline M_{0,n}$, which generalizes Keel's presentation of the Chow ring.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves an additive blowup formula for oriented cohomology theories in the non-A^1-invariant category of motivic spectra constructed by Annala-Hoyois-Iwasa. Specializing to A^1-invariant theories, it derives explicit presentations (generators and relations) for the oriented cohomology rings of blowups of smooth schemes along smooth centers, with concrete computations for del Pezzo surfaces, the blowup of P^3 along the twisted cubic, and the blowup of P^5 along the Veronese surface (identified with the moduli space of complete conics). It recovers classical enumerative results such as Steiner's 3264 conics using arbitrary oriented theories and concludes with a presentation of the oriented cohomology rings of Mbar_{0,n} that generalizes Keel's Chow ring presentation.

Significance. If the derivations hold, the work extends intersection-theoretic computations to a wider class of oriented cohomology theories beyond A^1-invariant ones, providing concrete ring presentations that recover and generalize classical results like Keel's. The explicit examples and recovery of enumerative geometry problems demonstrate practical utility. The non-A^1 setting and use of the Annala-Hoyois-Iwasa framework are novel contributions to motivic homotopy and algebraic geometry.

major comments (2)
  1. [Introduction and the section deriving the Mbar_{0,n} presentation] The abstract and introduction state that an additive blowup formula is proved in the non-A^1 category, followed by specialization yielding ring presentations for blowups and Mbar_{0,n}. However, an additive decomposition as modules does not automatically determine the ring multiplication or the precise ideal of relations among divisor classes (as in Keel's geometric intersections and linear equivalences). The manuscript must explicitly supply or cite the multiplicative structure (e.g., a projective bundle theorem or compatibility of products with the blowup formula) in the relevant section deriving the presentations; without this, the step from additivity to the claimed generators-and-relations for Mbar_{0,n} is not load-bearing.
  2. [Sections on explicit blowup presentations and the Mbar_{0,n} case] In the specialization to A^1-invariant theories and the explicit examples (del Pezzo, twisted cubic, Veronese), the relations in the ring presentations are asserted to generalize Keel's. The manuscript should verify that the intersection products and linear equivalences used to generate the relation ideal remain valid under the oriented cohomology operations in the Annala-Hoyois-Iwasa category; this needs a concrete check or reference to a theorem establishing multiplicativity.
minor comments (2)
  1. Notation for the oriented cohomology theory (e.g., the symbol for the theory and its grading) should be introduced uniformly at the first use and kept consistent across the blowup formula and the Mbar_{0,n} presentation.
  2. [Example of blowup of P^5 along Veronese surface] The abstract mentions recovery of Steiner's 3264 conics; the corresponding computation in the Veronese blowup example would benefit from a brief table or explicit generator count to make the enumerative application self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the transition from the additive blowup formula to the ring presentations requires more explicit justification. We will revise the manuscript to strengthen the exposition on multiplicative structure and compatibility with geometric relations.

read point-by-point responses

  1. Referee: [Introduction and the section deriving the Mbar_{0,n} presentation] The abstract and introduction state that an additive blowup formula is proved in the non-A^1 category, followed by specialization yielding ring presentations for blowups and Mbar_{0,n}. However, an additive decomposition as modules does not automatically determine the ring multiplication or the precise ideal of relations among divisor classes (as in Keel's geometric intersections and linear equivalences). The manuscript must explicitly supply or cite the multiplicative structure (e.g., a projective bundle theorem or compatibility of products with the blowup formula) in the relevant section deriving the presentations; without this, the step from additivity to the claimed generators-and-relations for Mbar_{0,n} is not load-bearing.

    Authors: We agree that an explicit discussion of the multiplicative structure is needed to make the specialization step fully rigorous. In the revised manuscript we will add a short subsection (following the statement of the additive blowup formula) that recalls the projective bundle theorem and the compatibility of products with blowups in the A^1-invariant case of the Annala-Hoyois-Iwasa framework. We will cite the relevant results on oriented cohomology rings (which guarantee that the ring structure descends under specialization) and explain how the ideal of relations is generated by the same geometric intersections and linear equivalences that appear in Keel's work, now interpreted in the oriented cohomology ring. This will render the derivation of the Mbar_{0,n} presentation load-bearing. revision: yes

  2. Referee: [Sections on explicit blowup presentations and the Mbar_{0,n} case] In the specialization to A^1-invariant theories and the explicit examples (del Pezzo, twisted cubic, Veronese), the relations in the ring presentations are asserted to generalize Keel's. The manuscript should verify that the intersection products and linear equivalences used to generate the relation ideal remain valid under the oriented cohomology operations in the Annala-Hoyois-Iwasa category; this needs a concrete check or reference to a theorem establishing multiplicativity.

    Authors: We concur that a concrete reference or verification strengthens the claim. In the revision we will insert a brief paragraph in the section on explicit blowup presentations stating that, upon specialization to A^1-invariant oriented cohomology theories, the intersection product coincides with the usual one on smooth schemes (by the axioms of oriented cohomology and the deformation-to-the-normal-cone construction, which is available in the A^1-invariant setting). We will add a reference to the compatibility theorems in the literature on A^1-invariant oriented theories and include a short explicit verification for the del Pezzo surface example, confirming that the relations among divisor classes are unchanged. The same argument applies verbatim to the Mbar_{0,n} case. revision: yes

Circularity Check

0 steps flagged

No circularity; central claims rest on external cited frameworks

full rationale

The paper proves an additive blowup formula in the Annala-Hoyois-Iwasa non-A1-invariant motivic spectra category (external construction) and then specializes to A1-invariant theories to obtain ring presentations that generalize Keel's known Chow ring result for Mbar_{0,n}. No step reduces a claimed prediction or theorem to a quantity defined or fitted inside the paper itself; all load-bearing inputs are independent prior results with no self-citation chains or definitional loops. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the framework of oriented cohomology theories in motivic spectra; no free parameters or new postulated entities are indicated in the abstract.

axioms (1)
  • domain assumption Existence and basic properties of oriented cohomology theories in the non-A1-invariant category of motivic spectra as constructed by Annala, Hoyois, and Iwasa
    The blowup formula is proved inside this external framework.

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Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    For a scheme S, the S-valued points are given by isomorphism classes of rπ:CÑS , tsi|i“ 1, ...,nu, σ: CÑXs , where C is a flat, proper family of at worst nodal singular projective, connected, reduced curves over S of arithmetic genus g, si are disjoint sections which map to the smooth locus of each fiber andσ ˚rCss “β,@sPSwhererC ssis the Chow fundamental...

    work page

  2. [2]

    (Stability condition) For everysPSand every irreducible componentEĂC s, we have • If E–P 1 and σ maps E to a point, then E must contain at least 3 special points (i.e., a marked point or a nodal point) • IfEhas arithmetic genus 1 andσmaps it to a point, thenEmust contain at least 1 special point. Remark B.2.The stability condition on the fiber is equivale...

    work page