[Paper Review] Rational Fibrations of $\bar{M}_{5,1}$ and $\bar{M}_{6,1}$

David Jensen|arXiv (Cornell University)|Dec 22, 2010

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper constructs rational fibrations from the moduli spaces $\overline{M}_{5,1}$ and $\overline{M}_{6,1}$ to lower-dimensional moduli spaces of pointed rational curves, using geometric constructions involving del Pezzo surfaces and osculating hyperplanes. The key result is that the Weierstrass divisor $BN^{1}_{5,(0,5)}$ and a new divisor $D_6$ on $\overline{M}_{6,1}$ generate extremal rays in the effective cone, providing new examples of extremal divisors via rational maps rather than birational contractions.

ABSTRACT

This is the second of two papers on the birational geometry of $\bar{M}_{g,1}$. We construct rational maps from $\bar{M}_{5,1}$ and $\bar{M}_{6,1}$ to lower-dimensional moduli spaces. As a consequence, we identify geometric divisors that generate extremal rays of the effective cones for these spaces.

Motivation & Objective

• To construct rational maps from $\overline{M}_{5,1}$ and $\overline{M}_{6,1}$ to simpler moduli spaces, overcoming the obstruction that no non-trivial morphisms exist to lower-dimensional varieties.
• To identify geometric divisors in $\overline{M}_{5,1}$ and $\overline{M}_{6,1}$ that generate extremal rays in the effective cone $\overline{NE}^1$.
• To extend the understanding of extremal rays beyond birational contractions by using rational maps induced by geometric structures such as del Pezzo surfaces and osculating hyperplanes.
• To provide a new geometric mechanism—via pullbacks of boundary divisors under rational maps—for detecting extremal rays in the effective cone of moduli spaces of curves.

Proposed method

• For $\overline{M}_{5,1}$, construct a rational map $\phi_5$ by using the canonical embedding of a genus 5 curve in $\mathbb{P}^4$ as a complete intersection of three quadrics, and analyzing the intersection of the osculating hyperplane with the associated del Pezzo surface.
• Blow up the del Pezzo surface at the marked point and then again at the intersection point of strict transforms, producing a $\mathbb{P}^1$ with four marked points, yielding a map to $\overline{M}_{0,4}$.
• For $\overline{M}_{6,1}$, use the fact that a general genus 6 curve embeds as a section of $|-2K_Y|$ in a smooth quintic del Pezzo surface $Y$, and define a rational map $\phi_6$ by forgetting the curve and keeping the marked point, mapping to $Y/S_5 \cong \widetilde{M}_{0,5}$.
• Analyze the images of pointed Brill-Noether divisors under $\phi_5$ and $\phi_6$, showing they lie in pullbacks of boundary divisors on $\overline{M}_{0,4}$ and $\overline{M}_{0,5}$.
• Use the fact that if two divisors map to the same point under a rational map, and one is not a multiple of the other, then both generate extremal rays in $\overline{NE}^1$ by Proposition 2.3.
• Leverage numerical equivalence and known divisor relations (e.g., $BN^{1}_{4,(0,3)} = BN^{1}_{5,(0,5)} + BN^{1}_{3}$) to rule out linear equivalence and confirm extremality.

Experimental results

Research questions

• RQ1 Can rational fibrations from $\overline{M}_{g,1}$ to moduli spaces of rational curves reveal extremal rays in the effective cone?
• RQ2 Does the Weierstrass divisor $BN^{1}_{5,(0,5)}$ generate an extremal ray in $\overline{NE}^1(\overline{M}_{5,1})$?
• RQ3 Is there a geometric divisor on $\overline{M}_{6,1}$, not of pointed Brill-Noether type, that generates an extremal ray?
• RQ4 Can the pullback of boundary divisors under rational maps detect extremal rays in the effective cone of $\overline{M}_{g,1}$?

Key findings

• The Weierstrass divisor $BN^{1}_{5,(0,5)}$ generates an extremal ray in $\overline{NE}^1(\overline{M}_{5,1})$, as shown by its image under $\phi_5$ being contained in the pullback of a point on $\overline{M}_{0,4}$.
• The pointed Brill-Noether divisor $BN^{1}_{4,(0,3)}$ is the pullback of an ample divisor on $\overline{M}_{0,4}$, which implies it generates an extremal ray.
• The divisor $D_6$ on $\overline{M}_{6,1}$, defined as the closure of curves with a $g^2_6$ and a $g^1_4$ ramified at the marked point, generates an extremal ray in $\overline{NE}^1(\overline{M}_{6,1})$.
• The divisor $BN^{2}_{6,(0,2,4)}$ is numerically equivalent to a multiple of $BN^{1}_{4,(0,3)}$, as it cannot be linearly equivalent to $BN^{1}_{5,(0,5)}$ or $BN^{1}_{3}$ due to their non-moving nature.
• The rational map $\phi_5: \overline{M}_{5,1} \dashrightarrow \overline{M}_{0,4}$ is constructed via blow-ups of a del Pezzo surface associated to the curve and marked point, yielding a geometric parameterization of the moduli space.
• The rational map $\phi_6: \overline{M}_{6,1} \dashrightarrow \widetilde{M}_{0,5}$ arises from the unique embedding of a genus 6 curve in a quintic del Pezzo surface, with the marked point determining the image point.