[Paper Review] Log canonical models for the moduli space of pointed stable rational curves

Han‐Bom Moon|arXiv (Cornell University)|Jan 6, 2011

Algebraic Geometry and Number Theory19 references3 citations

TL;DR

This paper proves that the log canonical model of the moduli space of pointed stable rational curves with divisor $K_{\overline{M}_{0,n}} + \sum a_i\psi_i$ is isomorphic to Hassett's moduli space of weighted pointed stable rational curves $\overline{M}_{0,\mathcal{A}}$, without assuming the F-conjecture. The proof uses push-pull of divisor classes and positivity arguments to show ampleness of the pushforward of the canonical divisor, establishing a birational model that matches Hassett's space directly.

ABSTRACT

We run Mori's program for the moduli space of pointed stable rational curves with divisor $K +\sum a_{i}ψ_{i}$. We prove that, without assuming the F-conjecture, the birational model for the pair is the Hassett's moduli space of weighted pointed stable rational curves, without any modification of weight coefficients.

Motivation & Objective

• To establish a birational model of $\overline{M}_{0,n}$ via Mori's program using the divisor $K_{\overline{M}_{0,n}} + \sum a_i\psi_i$.
• To prove that this model is isomorphic to Hassett's moduli space $\overline{M}_{0,\mathcal{A}}$ for any weight datum $\mathcal{A} = (a_1, \dots, a_n)$.
• To remove the assumption of the F-conjecture in prior results that linked log canonical models to weighted moduli spaces.
• To provide a moduli-theoretic interpretation of the log canonical model as $\overline{M}_{0,\mathcal{A}}$ via pushforward and ampleness arguments.

Proposed method

• Compute the pushforward and pullback of divisor classes on $\overline{M}_{0,n}$ and $\overline{M}_{0,\mathcal{A}}$ to relate $\Delta_{\mathcal{A}} = K_{\overline{M}_{0,n}} + \sum a_i\psi_i$ to the reduction morphism $\varphi_{\mathcal{A}}$.
• Show that $\Delta_{\mathcal{A}} - \varphi_{\mathcal{A}}^*\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ is effective and supported on the exceptional locus of $\varphi_{\mathcal{A}}$.
• Use [Deb01, Lemma 7.11] to identify global sections of $\mathcal{O}(l\Delta_{\mathcal{A}})$ on $\overline{M}_{0,n}$ with those on $\overline{M}_{0,\mathcal{A}}$.
• Prove that $\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ is ample on $\overline{M}_{0,\mathcal{A}}$ by showing it intersects all curves non-negatively and is a limit of nef divisors.
• Apply Fedorchuk’s positivity result and induction on dimension to establish nefness of $\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$.
• Use the fact that $N^1(\overline{M}_{0,\mathcal{A}})$ is generated by boundary divisors to conclude ampleness from non-negative intersection with all curves.

Experimental results

Research questions

• RQ1 Is the log canonical model of $\overline{M}_{0,n}$ with divisor $K_{\overline{M}_{0,n}} + \sum a_i\psi_i$ isomorphic to Hassett’s moduli space $\overline{M}_{0,\mathcal{A}}$ without assuming the F-conjecture?
• RQ2 Can the pushforward $\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ be shown to be ample on $\overline{M}_{0,\mathcal{A}}$ using divisor push-pull and positivity techniques?
• RQ3 Does the log canonical model construction via $\mathrm{Proj}\big(\bigoplus H^0(\overline{M}_{0,n}, \mathcal{O}(l\Delta_{\mathcal{A}}))\big)$ yield $\overline{M}_{0,\mathcal{A}}$ as a moduli space?
• RQ4 Can the ampleness of $\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ be established even when $\varphi_{\mathcal{A}}^*\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ is not log canonical?

Key findings

• The log canonical model $\overline{M}_{0,n}(K_{\overline{M}_{0,n}} + \sum a_i\psi_i)$ is isomorphic to $\overline{M}_{0,\mathcal{A}}$ for any weight datum $\mathcal{A} = (a_1, \dots, a_n)$, without assuming the F-conjecture.
• The pushforward $\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ is ample on $\overline{M}_{0,\mathcal{A}}$, which implies the isomorphism of the log canonical models.
• The proof relies on showing that $\Delta_{\mathcal{A}} - \varphi_{\mathcal{A}}^*\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ is effective and supported on the exceptional locus, so sections of $\mathcal{O}(l\Delta_{\mathcal{A}})$ on $\overline{M}_{0,n}$ correspond to those on $\overline{M}_{0,\mathcal{A}}$.
• The ampleness of $\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}})$ is established via induction and positivity: it intersects all curves non-negatively and remains nef under small perturbations by boundary divisors.
• A global positive lower bound $\epsilon_{\mathcal{A}}$ exists for the intersection number $\varphi_{\mathcal{A}*}(\Delta_{\mathcal{A}}) \cdot B$ over all irreducible curves $B$ on $\overline{M}_{0,\mathcal{A}}$.
• The result generalizes Simpson’s theorem (Theorem 1.2) to non-symmetric weights, and the symmetric case recovers the known result via the identity $K_{\overline{M}_{0,n}} + \alpha\psi = (1+\alpha)(K_{\overline{M}_{0,n}} + \frac{2\alpha}{1+\alpha}D)$.