Log algebraic hyperbolicity of $\overline{M}_{0,n}$

TL;DR

The paper proves that the moduli space $\overline{M}_{0,n}$ of stable $n$-pointed rational curves with its boundary $\Delta$ is algebraically hyperbolic. It reduces the problem to two birational models, $((\mathbb{P}^1)^{n-3},D)$ and $(\mathbb{P}^{n-3},H)$, establishing explicit hyperbolicity bounds $\epsilon=\frac{1}{(n-3)^2}$ for $n\ge 4$ (and trivial cases for small $n$) via blow-up preservation of log hyperbolicity. The two model pairs are shown to be algebraically hyperbolic using a combination of log-Riemann–Hurwitz for projections and hyperplane-imbedding arguments, which then transfer to $$(\overline{M}_{0,n},\Delta)$$ through Keel and Kapranov's blow-up structures. This yields the concrete inequality $2g(C)-2+i(C,\Delta) \ge \frac{1}{(n-3)^2}\deg_{K+\Delta} C$ for non-boundary curves, implying strong hyperbolicity properties and constraining the geometry of curves on the moduli space.

Abstract

We show that the moduli space of stable n-pointed rational curves $\overline{M}_{0,n}$ with its boundary $Δ$ is algebraically hyperbolic.

Theorems & Definitions (28)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3: keel2005equationsbarm0n, Theorem 1.1
• Theorem 2.4: hassett2002modulispacesweightedpointed, section 6.3
• Theorem 2.5: hassett2002modulispacesweightedpointed, section 6.2
• Definition 2.6: chen2001algebraichyperbolicitylogvarieties
• Proposition 3.1: chen2001algebraichyperbolicitylogsurfaces
• proof