[Paper Review] Towards the ample cone of $\mgn$

Angela Gibney, Seán Keel+1 more|ArXiv.org|Jun 27, 2000

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper proposes a conjectural characterization of the ample cone of the moduli space $\overline{M}_{g,n}$ of stable $n$-pointed genus $g$ curves: a divisor is ample if and only if it intersects all 1-dimensional strata positively. The key result shows that this conjecture holds for all $g$ if and only if it holds for $g=0$, reducing the general problem to the case of genus zero, where it is known to be true in low genera and supported by evidence from GIT quotients and toric contractions.

ABSTRACT

In this paper we study the ample cone of the moduli space $\mgn$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\mgn$ is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More precisely, there is a natural finite map $r: \vmgn 0. 2g+n. o \mgn$ whose image is the locus $ gn$ of curves with all components rational. Any 1-strata either lies in $ gn$ or is numerically equivalent to a family $E$ of elliptic tails and we show that a divisor $D$ is nef iff $D \cdot E \geq 0$ and $r^*(D)$ is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of $\mgn$ for $g \geq 1$ showing that any fibration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary. Finally, by more ad-hoc arguments, we prove the nefness of certain special classes.

Motivation & Objective

• To resolve Mumford's open question on the structure of the ample cone of $\overline{M}_{g,n}$.
• To provide a conjectural linear inequality description of the ample cone based on positivity against 1-dimensional strata.
• To reduce the general problem for $g \geq 1$ to the genus zero case via a finite map $r: \overline{M}_{0,2g+n} \to \overline{M}_{g,n}$.
• To analyze contractions of $\overline{M}_{g,n}$, particularly fibrations and divisorial contractions, and show they factor through tautological maps or blow down elliptic tails.
• To establish that the Mori cone is generated by rational curves and elliptic tails, with the latter being the only exceptional locus for relative Picard number one divisorial contractions when $g \geq 5$.

Proposed method

• Use a natural finite morphism $r: \overline{M}_{0,2g+n} \to \overline{M}_{g,n}$ whose image is the locus $\overline{R}_{g,n}$ of rational curves, to relate nefness on $\overline{M}_{g,n}$ to nefness on $\overline{M}_{0,2g+n}$.
• Show that a divisor $D$ on $\overline{M}_{g,n}$ is nef if and only if it has non-negative intersection with all 1-strata and $D|_{\overline{F}_{g,n}}$ is nef, where $\overline{F}_{g,n}$ is the locus of flag curves.
• Characterize the Mori cone $\overline{NE}_1(\overline{M}_{g,n})$ as a quotient of $\overline{NE}_1(\overline{M}_{0,2g+n}) \times \mathbb{R}_{\geq 0}$, with generators from rational curves and elliptic tails.
• Analyze fibrations of $\overline{M}_{g,n}$ for $g \geq 2$, proving they factor through tautological fibrations (forgetting points) composed with birational morphisms.
• Use deformation theory and relative Néron-Severi groups to show that for $g \geq 5$, the only relative Picard number one divisorial contraction is the blowdown of elliptic tails.
• Leverage known results on $\overline{M}_{0,n}$, including Kapranov's inverse limit description and the Losev-Manin toric compactification $\overline{L}_{n-2}$, to provide evidence for the conjecture.

Experimental results

Research questions

• RQ1 Is the ample cone of $\overline{M}_{g,n}$ characterized by positivity against all 1-dimensional strata?
• RQ2 Does the conjecture that a divisor is ample iff it intersects all 1-strata positively hold for $g \geq 1$ if and only if it holds for $g = 0$?
• RQ3 What are the possible fibrations and divisorial contractions of $\overline{M}_{g,n}$ for $g \geq 2$?
• RQ4 Are the only extremal rays in $\overline{NE}_1(\overline{M}_{g,n})$ generated by rational curves and elliptic tails?
• RQ5 Can the Mori cone of $\overline{M}_{g,n}$ be described as a quotient of the Mori cone of $\overline{M}_{0,2g+n}$?

Key findings

• The conjecture $F_1(\overline{M}_{g,n})$ holds if and only if it holds for $g = 0$, reducing the general problem to the genus zero case.
• A divisor $D$ on $\overline{M}_{g,n}$ is nef if and only if it has non-negative intersection with all 1-strata and $D|_{\overline{F}_{g,n}}$ is nef, where $\overline{F}_{g,n}$ is the flag curve locus.
• The Mori cone $\overline{NE}_1(\overline{M}_{g,n})$ is generated by rational curves (in $\overline{R}_{g,n}$) and elliptic tails, with the latter being numerically equivalent across families.
• For $g \geq 5$ and characteristic zero, the only relative Picard number one divisorial contraction of $\overline{M}_{g,n}$ is the blowdown of the elliptic tails locus.
• The conjecture $F_1(\overline{M}_{0,n})$ is supported by evidence from toric compactifications $\overline{L}_{n-2}$, which are contractions of $\overline{M}_{0,n}$ and satisfy $F_k(\overline{L}_{n-2})$ for all $k$.
• In characteristic zero, $F_1(\overline{M}_{g,n})$ holds for $g+n \leq 7$ and when $n=0$ for $g \leq 11$, extending previous results for $g \leq 4$.