[Paper Review] A perfect stratification of M_g for g at most 5

Claudio Fontanari, Eduard Looijenga|ArXiv.org|Aug 25, 2007

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper constructs a perfect stratification of the moduli space $\mathcal{M}_g$ for $g \leq 5$ by analyzing the geometry of quadric systems through canonical curves. It proves that the strata are affine and their closures form a $\mathbb{Q}$-basis for the Chow ring, yielding a new, simpler proof that $A^\bullet(\mathcal{M}_g) = \mathbb{Q}[\lambda]/(\lambda^{g-1})$, confirming the Chow ring is generated by the $\lambda$-class in this range.

ABSTRACT

We find for g at most 5 a stratification of depth g-2 of the moduli space of curves M_g with the property that its strata are affine and the classes of their closures provide a Q-basis for the Chow ring of M_g. The first property confirms a conjecture of one of us. The way we establish the second property yields new (and simpler) proofs of theorems of Faber and Izadi which, taken together, amount to the statement that in this range the Chow ring is generated by the lambda-class.

Motivation & Objective

• To construct an affine stratification of $\mathcal{M}_g$ of depth $g-2$ for $g \leq 5$, confirming a conjecture on the existence of such a stratification.
• To show that the closures of the strata form a $\mathbb{Q}$-basis for the Chow ring $A^\bullet(\mathcal{M}_g)$, establishing a 'perfect' stratification.
• To provide a new, simplified proof that the tautological Chow ring of $\mathcal{M}_g$ for $g \leq 5$ is generated by the $\lambda$-class, i.e., $A^\bullet(\mathcal{M}_g) = \mathbb{Q}[\lambda]/(\lambda^{g-1})$.
• To demonstrate that the open stratum defined by curves with no effective even theta characteristic is affine, serving as the top stratum in the construction.

Proposed method

• The authors use the linear system of quadrics passing through the canonical image of a genus $g$ curve to define strata based on the dimension of this system.
• For $g=3,4,5$, they define strata by geometric conditions: absence of even theta characteristics (codimension 1), non-trigonal curves (codimension 2), and plane quintics with singularities (codimension 3).
• They prove that each stratum is affine by showing the complement of the stratum is a finite morphism from a weighted projective space minus a hypersurface, leveraging Lemma 2.2 on affine quotients.
• The Chow ring of each stratum is trivial, and the restriction of the $\lambda$-class to each stratum vanishes in successive differences, implying $\lambda^k$ is proportional to the class of the $k$-th stratum.
• They use the fact that $A^1(\mathcal{M}_g \setminus \mathcal{T}_g) = 0$ and that $\mathcal{R}^\bullet(\mathcal{M}_g \setminus \mathcal{T}_g)$ is generated by $\kappa_2$, which vanishes in the Chow ring due to the vanishing of $\lambda$, to conclude the Chow ring is $\mathbb{Q}$.
• The proof relies on known results from Mumford, Faber, Izadi, and Teixidor i Bigas, combined with new geometric arguments on canonical curves and their quadric systems.

Experimental results

Research questions

• RQ1 Does there exist an affine stratification of $\mathcal{M}_g$ of depth $g-2$ for $g \leq 5$?
• RQ2 Can the closures of the strata in such a stratification form a $\mathbb{Q}$-basis for the Chow ring $A^\bullet(\mathcal{M}_g)$?
• RQ3 Is the Chow ring of $\mathcal{M}_g$ for $g \leq 5$ generated by the $\lambda$-class, and can this be proven independently of prior work?
• RQ4 What is the geometric meaning of the vanishing of $\lambda$ in the Chow ring of the open stratum $\mathcal{M}_g \setminus \mathcal{T}_g$?
• RQ5 Can the tautological Chow ring of $\mathcal{M}_g$ be fully described via a filtration where each successive difference has trivial Chow ring and $\lambda$-class vanishes?

Key findings

• For $g \leq 5$, the moduli space $\mathcal{M}_g$ admits a perfect stratification of depth $g-2$ with all strata affine.
• The closures of the strata in this stratification form a $\mathbb{Q}$-basis for the Chow ring $A^\bullet(\mathcal{M}_g)$.
• The $k$-th stratum's closure represents a non-zero multiple of $\lambda^k$ in $A^\bullet(\mathcal{M}_g)$, so $\lambda^k$ is non-zero for $k \leq g-2$ and $\lambda^{g-1} = 0$.
• The Chow ring $A^\bullet(\mathcal{M}_g)$ is isomorphic to $\mathbb{Q}[\lambda]/(\lambda^{g-1})$ for $g \leq 5$, and this is proven via a new, simpler argument independent of Faber and Izadi's original proofs.
• The open stratum $\mathcal{M}_g \setminus \mathcal{T}_g$, consisting of curves with no effective even theta characteristic, is affine and has trivial Chow ring.
• For $g=5$, two distinct perfect stratifications are constructed: one using the locus $\mathcal{T}_5$ of curves with a point $p$ with $h^0(3p) \geq 2$, and another using $\mathcal{M}_5'$, both yielding the same Chow ring basis.