A cellular (co)homology computation for $\overline{M_{0,n}}$

Abstract: In this article we set up and showcase cellular computations for (co)homology with values in strictly $\mathbb{A}<sup>1$-invariant sheaves. These computations encapsulate many classical invariants like Chow groups and singular cohomology of the real points. They also extend enumerative arguments from algebraically closed fields to more general fields. The spaces considered here have to admit a cellular structure. Instead of using the classical notion of cellularity, i.e. having a stratification by affine spaces, more general stratifications by cohomologically trivial spaces are used, following Morel--Sawant. Examples of cellular spaces include projective spaces and their products, but also spaces such as $\overline{M_{0,n}}$, the moduli space of stable genus $0$ curves with $n$ marked points. For these examples, we showcase the computations and show how to derive the classical results. Hopefully, the following text provides enough evidence to be convincing that such computations are doable and is encouraging to start computing the cohomology for more cellular spaces. This is part of the author's PhD thesis.

• The paper introduces a cellular (co)homology framework for moduli spaces using cohomologically trivial cells.
• It employs explicit residue maps and matrix computations to reduce complex cohomology and Chow group calculations over arbitrary fields.
• The method refines classical enumerative geometry by integrating Milnor–Witt and quadratic invariants with arithmetic sensitivity.

Cellular (Co)homology Computation for $\overline{M_{0,n}}$

Introduction

This work develops explicit methods for computing (co)homology theories, such as Milnor-Witt and Chow groups, for moduli stacks $\overline{M_{0,n}}$ of stable genus $0$ curves with $n$ marked points. The central approach leverages cellular stratifications—generalizing classical topological cellular decompositions to algebraic varieties—incorporating the Morel–Sawant notion of cohomologically trivial cells, thereby surpassing strict affine cell decompositions. This framework supports computations over arbitrary base fields and with coefficients in strictly $\mathbb{A}^1$-invariant sheaves, encompassing K-theoretic and quadratic refinements of classical invariants, with substantial relevance for arithmetic and enumerative geometry.

Mathematical Setting and Motivation

The paper unifies classical topological ideas—CW complexes and cellular (co)homology—with the modern requirements of algebraic geometry over general fields and with enhanced coefficients. Strictly cellular schemes are rare in algebraic geometry beyond projective spaces, hence the adoption of stratifications by cohomologically trivial subvarieties. Such cells share the vanishing higher Nisnevich cohomology property with affine spaces, enabling effective cohomological reduction arguments. The methodology allows explicit computation of Chow groups, cohomology with coefficients in Milnor and Milnor-Witt K-theory, and quadratic invariants such as the Grothendieck-Witt ring.

Enumerative geometric problems, exemplified by counts of rational curves passing through prescribed points or lines on cubic surfaces, classically utilize Chow group computations over algebraically closed fields. The work here extends such computations, via Milnor-Witt tools, to arbitrary base fields, thus furnishing refined invariants (e.g., signed counts in the Grothendieck-Witt group) that incorporate arithmetic information beyond mere cardinality.

The moduli spaces $\overline{M_{0,n}}$ exhibit recursive and combinatorial structures, with boundary stratifications indexed by subsets and explicit recurrence via blow-ups and product structures. This structure underpins the tractability of the proposed cellular approach.

Cellular Structures and Cohomological Triviality

The crucial technical innovation is allowing cell decompositions by smooth, affine, cohomologically trivial varieties (as in Morel–Sawant [MorelSawant]), not just affine spaces. For example, complements of arrangements of hyperplanes in affine space (such as configuration spaces for moduli) satisfy this property. Over such a filtration of a scheme $X$, the cohomology (or Chow–Witt group) computation reduces to a chain complex whose terms are values of the coefficient sheaf on the cells, and whose differentials are explicit via residue maps as formalized in the Rost–Schmid complex.

The cohomological machinery is robust with respect to line bundle twists (important for refined invariants and cycle class maps), and supports recursive reductions to lower-dimensional cases, crucial for inductive computation on moduli spaces.

Computation: General Method and Key Examples

The general recipe for (co)homology computation involves:

1. Cellular Filtration: Identify a filtration of $X$ by cohomologically trivial strata.
2. Cell Cohomology: Compute $H^0(\text{cell},\underline{M})$ for the sheaf $\underline{M}$; for strictly cellular cells, these reduce to sections over the base field.
3. Residue Maps and Differentials: Employ geometric curves within cells and their intersection with boundaries to deduce explicit formulas for the differentials in the chain complex.
4. Matrix Computation: Explicitly write differential matrices; homology computes the desired (co)homology.

Projective Spaces and Products: Using strict cell decompositions, the cohomology with Milnor-Witt coefficients is given explicitly, revealing failures of projective bundle formulas and giving explicit spectral sequences that recover classical, Chow, and quadratic invariants simultaneously.

Complements of Hyperplane Arrangements: These form the cells in the stratification of (open subsets of) $\overline{M_{0,n}}$0; the paper proves their cohomological triviality and provides explicit counts of the summands appearing in $\overline{M_{0,n}}$1 via combinatorics of the arrangement lattice.

$\overline{M_{0,n}}$2 Case: As an explicit illustration, the chain complex for $\overline{M_{0,n}}$3 is written explicitly in terms of Milnor-Witt $\overline{M_{0,n}}$4-theory of the base field, and the differential matrices are determined. The calculations reveal the precise nature and arithmetic sensitivity of the (co)homology, with dimensions and torsion structure reflecting deep geometric and arithmetic content.

Structural Results and Conjectures

The computational analysis leads to a conjectural structure theorem for the (co)homology groups $\overline{M_{0,n}}$5 for all $\overline{M_{0,n}}$6 and twists $\overline{M_{0,n}}$7, asserting a canonical decomposition into summands isomorphic to $\overline{M_{0,n}}$8, its kernel under multiplication by $\overline{M_{0,n}}$9, and the corresponding cokernel, with ranks determined by the classical (Chow, singular cohomology) realizations of the space and the rank of their $0$0-torsion. This conjecture, if substantiated, would articulate the precise relationship between quadratic refined invariants and their classical counterparts, and guide the extension of enumerative results to fields with richer structure.

The analysis also underscores the algorithmic tractability: the differentials reduce to residue computations along rational curves, which, in the recursive geometry of moduli spaces, can always be traced down to explicit residue and twist data, albeit with rapidly increasing combinatorial complexity as $0$1 grows.

Implications and Future Directions

The presented method systematically opens strictly cellular computations to vast new classes of moduli spaces. The explicit approach yields results in arbitrary characteristic (excluding $0$2 for technical reasons) and for all twists by line bundles. This leads to refined arithmetic enumerative results and supplies foundational data for motivic, quadratic, and $0$3-homotopy-theoretic invariants of moduli spaces.

The approach generalizes to a wide array of moduli spaces and varieties admitting a sufficiently compatible decomposition, indicating broad applicability in the study of motivic phenomena, refined enumerative geometry, quadratic refinements, and algebraic cycles.

Future work will likely involve:

• Extending computations to higher $0$4, leveraging computer-assisted matrix calculations.
• Elucidating geometric and operadic interpretations of generators, relations, and ring structures in Chow-Witt rings of moduli.
• Applying these tools to resolve questions around refined counts, ranks, and symmetries in enumerative arithmetic geometry.
• Investigating implications for $0$5-homotopy types, quadratic invariants of moduli operads, and real cycle class maps.

Conclusion

The development of cellular homology and cohomology computations using cohomologically trivial cells provides a powerful and flexible tool for understanding the refined (co)homology of moduli spaces such as $0$6. This framework unifies classical and arithmetic invariants, deepens the interplay between enumerative geometry and quadratic refinement, and opens new avenues for explicit computation across algebraic geometry and motivic homotopy theory. The arithmetic and geometric complexity is encoded in a combinatorial–homological package, transparently accessible via explicit chain complexes and residue computations, and poised for further theoretical exploration and computational expansion.