[Paper Review] Strata of abelian differentials and the Teichmueller dynamics

Dawei Chen|arXiv (Cornell University)|Nov 24, 2012

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes an algebraic framework to study strata of abelian differentials and their Teichmüller dynamics using intersection theory on moduli spaces of curves. It computes the cycle class of the stratum $¯{\mathcal{H}}(2,1^{2g-4})$ via the Porteous formula, proves extremality of several pointed Brill–Noether divisors in low genus using Lyapunov exponents of Teichmüller curves, and provides an algebraic alternative to analytic methods for divisor class calculations.

ABSTRACT

This paper focuses on the interplay between the intersection theory and the Teichmueller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a non-simple zero, and verify a number of extremal effective divisors on the moduli space of pointed curves in low genus.

Motivation & Objective

• To develop an algebraic approach to compute cycle classes of strata of abelian differentials in the moduli space of curves.
• To verify the extremality of pointed Brill–Noether divisors in low genus using Teichmüller dynamics and Lyapunov exponents.
• To provide an algebraic alternative to analytic methods—such as the Tau function—for computing divisor classes on compactified Hodge bundles.
• To understand the birational geometry of the projectivized Hodge bundle $\mathbb{P}\overline{\mathcal{H}}$ by analyzing the position of the divisor $\mathbb{P}\overline{\mathcal{H}}(2,1^{2g-4})$ in the pseudo-effective cone.
• To establish a bridge between Teichmüller dynamics and algebraic geometry by linking dynamical invariants (Lyapunov exponents) to effective divisor theory.

Proposed method

• Use the Porteous formula to compute the cycle class of the stratum $\mathbb{P}\mathcal{H}(\mu)$ in the Chow ring of $\mathcal{M}_{g,n}$.
• Lift Teichmüller curves from $\mathcal{H}(\mu)$ to $\overline{\mathcal{M}}_{g,n}$ and compute their intersection numbers with divisor classes.
• Apply the formula $\frac{C \cdot BN^{1}_{g,\underline{a}}}{C \cdot \lambda} = -1 + \sum_{i=1}^{n} \frac{a_i}{2L}$ to relate Lyapunov exponents $L$ to intersection numbers.
• Use the slope bound $s(C) \leq 8 + \frac{4}{g}$ to control the ratio of intersection numbers and derive uniform bounds.
• Construct ample divisors $D_{\underline{s}}$ with small coefficients $s_i$ to establish extremality via comparison of intersection numbers.
• Leverage known values of Lyapunov exponent sums $L_{(\mu)}$ for strata $\mathcal{H}(\mu)$ to infer the existence of Teichmüller curves with $L > 2 + \epsilon$.

Experimental results

Research questions

• RQ1 What is the cycle class of the stratum $\mathbb{P}\overline{\mathcal{H}}(2,1^{2g-4})$ in the Chow ring of $\overline{\mathcal{M}}_{g,2g-2}$?
• RQ2 Can the extremality of pointed Brill–Noether divisors on $\overline{\mathcal{M}}_{g,n}$ be established using Teichmüller dynamics and intersection theory?
• RQ3 How does the limit of Lyapunov exponent sums along Teichmüller curves relate to the extremality of effective divisors?
• RQ4 Is the divisor $\mathbb{P}\overline{\mathcal{H}}(2,1^{2g-4})$ on the boundary of the pseudo-effective cone of $\mathbb{P}\overline{\mathcal{H}}$?
• RQ5 Can algebraic methods replace analytic techniques—such as the Tau function—for computing divisor classes in the Hodge bundle?

Key findings

• The cycle class of $\mathbb{P}\overline{\mathcal{H}}(2,1^{2g-4})$ is computed algebraically via the Porteous formula, providing a new method independent of analytic approaches.
• The divisor $\mathbb{P}\overline{\mathcal{H}}(2,1^{2g-4})$ lies on the boundary of the pseudo-effective cone of $\mathbb{P}\overline{\mathcal{H}}$, confirming its geometric significance.
• For $2 \leq g \leq 4$, the Weierstrass divisor $W = BN^{1}_{g,(g)}$ is extremal in $\overline{\mathcal{M}}_{g,1}$, with $L > 1$ for $g=2$, $L > 1.5$ for $g=3$, and $L > 2$ for $g=4$.
• The pointed Brill–Noether divisors $BN^{1}_{2,(1,1)}$, $BN^{1}_{3,(2,1)}$, $BN^{1}_{4,(3,1)}$, $BN^{1}_{4,(2,2)}$, $BN^{1}_{3,(1^3)}$, $BN^{1}_{4,(2,1,1)}$, $BN^{1}_{4,(1^4)}$, and $BN^{1}_{5,(1^5)}$ are all extremal in their respective moduli spaces.
• The Lyapunov exponent sum $L_{(\mu)}$ for strata $\mathcal{H}(\mu)$ is used to infer the existence of Teichmüller curves with $L > 2 + \epsilon$, enabling extremality proofs via intersection bounds.
• The method provides a uniform bound $\frac{C \cdot BN^{1}_{g,\underline{a}}}{C \cdot D_{\underline{s}}} \leq -d$ for some $d > 0$, independent of the curve $C$, by choosing $D_{\underline{s}}$ ample with small coefficients $s_i$.