An introduction to the intersection theory of the moduli space of curves

Technical Summary: An Introduction to the Intersection Theory of the Moduli Space of Curves

Problem Statement
The central problem addressed in this survey is the understanding of the intersection theory of the moduli space of curves, $M_g$, specifically regarding the structure of its Chow ring, $A^*(M_g)$. The author investigates the relationship between the full Chow ring and the tautological ring, $R^*(M_g)$, a subring generated by classes naturally arising from the geometry of the universal family. The survey focuses on two fundamental questions:

1. Generation: Is the Chow ring $A^*(M_g)$ generated by tautological classes?
2. Relations: What are the complete set of relations among tautological classes?

The paper highlights that while the Chow ring is well-understood for low genera, the situation becomes increasingly complex as the genus $g$ grows, with the existence of non-tautological classes in higher genera remaining a significant open area of research.

Methodology and Framework
The paper employs the standard machinery of algebraic geometry, specifically intersection theory and Chow rings, adapted to the context of moduli spaces (often treated as Deligne–Mumford stacks).

• Foundational Tools: The survey reviews the definition of the Chow group $A_*(X)$ as cycles modulo rational equivalence. It utilizes proper pushforward ($f_*$) and flat pullback ($f^*$) maps, as well as the excision sequence for open/closed subschemes. The push-pull formula and self-intersection formulas (via excess intersection) are central to computing intersections.
• Tautological Classes: Following Mumford's philosophy, tautological classes are defined as those arising from the universal family. For $M_g$, these include:
  • $\kappa$-classes: Defined via the pushforward of powers of the first Chern class of the relative dualizing sheaf $\omega_f$ along the universal curve $f: C_g \to M_g$.
  • $\lambda$-classes: Defined as the Chern classes of the Hodge bundle $E = f_*\omega_f$.
  • $\psi$-classes: Defined on pointed moduli spaces $M_{g,n}$ via the sections of the universal curve.
• Stratification and Gluing: The methodology relies heavily on stratifying moduli spaces (e.g., by gonality or via stable graphs for compactifications) and using excision sequences to relate the Chow rings of strata to the whole space. For compactifications $\overline{M}_{g,n}$, the tautological ring is defined as the smallest system of subrings closed under pushforwards along gluing maps (associated with stable graphs) and forgetful maps.
• Computational Techniques: The paper discusses the use of Porteous formulas for degeneracy loci, Grothendieck–Riemann–Roch for relating Chern classes of pushforwards, and Künneth decompositions in cohomology to detect non-tautological classes.

Key Contributions and Results

1. Survey of the Tautological Ring:

   • The paper surveys the Faber conjectures regarding the structure of $R^*(M_g)$, including the generation by $\kappa$-classes up to degree $\lfloor g/3 \rfloor$, vanishing in degrees $> g-2$, and the existence of a perfect pairing in the top degree.
   • It notes that while the Faber–Zagier relations (conjectured by Faber and Zagier and proven by Pixton and Pandharipane) provide a large set of relations, it remains an open question whether they constitute all relations. Recent computational evidence (Canning) suggests the pairing in degree $g-2$ is not perfect for $g=24$, implying either the Faber conjecture (part 3) is false or the Faber–Zagier relations are incomplete.

2. Generation Results (When $A^*(M_g) = R^*(M_g)$):

   • The paper confirms that for $2 \le g \le 9$, the Chow ring is generated by tautological classes ($A^*(M_g) = R^*(M_g)$). This result synthesizes work by Mumford, Faber, Izadi, Penev–Vakil, and Canning–Larson.
   • Case Study: Genus 3: The author provides a detailed proof sketch for $g=3$. The strategy involves stratifying $M_3$ into the hyperelliptic locus $H$ and the locus of smooth plane quartics $P$.
     • $H$ is shown to have a trivial Chow ring (rationally) via its relation to $M_{0,8}/S_8$.
     • $P$ is analyzed as a quotient stack $U/GL_3$, showing its Chow ring is generated by Chern classes of the Hodge bundle (which are tautological).
     • Using the excision sequence, the proof reduces to showing the fundamental class $[H]$ is tautological. This is achieved by realizing $H$ as a degeneracy locus of a map between vector bundles on $C_2$ and applying the Porteous formula.

3. Non-Tautological Classes (When $A^*(M_g) \neq R^*(M_g)$):

   • The paper establishes that for $g=12$ and $g \ge 16$, the Chow ring is not generated by tautological classes.
   • Case Study: Genus 12: The author details the construction of a non-tautological class by Van Zelm. The class in question is the fundamental class of the bielliptic locus $B_{12}$ (curves admitting a degree 2 map to an elliptic curve).
     • The proof utilizes the gluing map $i: M_{1,11} \times M_{1,11} \to M_{12}$.
     • By pulling back the class $[B_{12}]$ via $i$, one obtains a class involving the diagonal $\Delta$ in $M_{1,11} \times M_{1,11}$.
     • The argument relies on Hodge theory and Künneth components. Specifically, $H^{11,0}(M_{1,11}) \neq 0$. The diagonal class has a non-zero component in $H^{11,0} \otimes H^{0,11}$.
     • Classes supported on the boundary of $M_{12}$ (and their pullbacks) are shown to lack this specific Hodge type. Since tautological classes pulled back via gluing maps must respect certain Hodge constraints (specifically, they cannot generate the specific odd-degree Hodge type found in the diagonal), $[B_{12}]$ cannot be tautological.

Significance and Claims
The paper positions itself as an introduction to the current state of the art in the intersection theory of $M_g$. Its significance lies in:

• Unifying Framework: It connects classical examples (Grassmannians, classifying stacks $BGL_k$) to the modern theory of moduli spaces of curves, illustrating how "tautological" classes arise naturally from universal properties.
• Clarifying the Boundary of Knowledge: It clearly delineates the known cases where the Chow ring is tautological ($g \le 9$) versus where it is not ($g=12, g \ge 16$), highlighting the specific genera ($10, 11, 13, 14, 15$) where the question remains open.
• Methodological Demonstration: By sketching proofs for $g=3$ (generation) and $g=12$ (non-generation), the paper demonstrates the specific technical tools required for these problems: stratification and excision for generation, and Hodge-theoretic Künneth analysis for non-generation.

The author modestly claims that while the integral Chow ring is only known for $g=2$, the rational Chow ring is understood for $g \le 9$, and the existence of non-tautological classes is established for specific higher genera. The paper does not claim to resolve the open cases but rather to provide the "flavor of the state-of-the-art tools" used to address them.