[Paper Review] Moduli of Trigonal Curves

Zvezdelina E. Stankova-Frenkel|arXiv (Cornell University)|Oct 12, 1997

Advanced Numerical Analysis Techniques19 references55 citations

TL;DR

This paper establishes the sharp upper bound $\frac{36(g+1)}{5g+1}$ for the slope $\delta_B/\lambda_B$ of trigonal fibrations over a base curve $B$, proving it is achieved precisely when all fibers are irreducible and a certain divisor class $\eta$ is numerically trivial. It further links this bound to the geometry of the Maroni locus in the trigonal locus $\overline{\mathfrak{T}}_g$, and computes the rational Picard group of $\overline{\mathfrak{T}}_g$ in even genus via Bogomolov semistability of an associated rank two vector bundle.

ABSTRACT

We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any fibration $X o B$ of stable curves with smooth general member is the ratio $δ_B/λ_B$ of the restrictions of the boundary class $δ$ and the Hodge class $λ$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base $B$. We associate to a trigonal family $X$ a canonical rank two vector bundle $V$, and show that for Bogomolov-semistable $V$ the slope satisfies the stronger inequality ${δ_B}/{λ_B}\leq 7+{6}/{g}$. We further describe the rational Picard group of the {trigonal} locus $\bar{\mathfrak T}_g$ in the moduli space $\bar{\mathfrak{M}}_g$ of genus $g$ curves. In the even genus case, we interpret the above Bogomolov semistability condition in terms of the so-called Maroni divisor in $\bar{\mathfrak T}_g$.

Motivation & Objective

• To determine the exact upper bound for the slope $\delta_B/\lambda_B$ of non-isotrivial trigonal fibrations over a base curve $B$.
• To characterize the families achieving this maximal slope in terms of geometric and cohomological conditions on the fibration and its associated vector bundle.
• To describe the rational Picard group of the trigonal locus $\overline{\mathfrak{T}}_g$ in $\overline{\mathfrak{M}}_g$, particularly in the even genus case.
• To interpret the Bogomolov semistability condition of the canonical rank two vector bundle $V$ associated to a trigonal family in terms of the Maroni divisor in $\overline{\mathfrak{T}}_g$.

Proposed method

• Construct a canonical rank two vector bundle $V$ associated to any trigonal fibration $X \to B$, which encodes the linear series $g^1_3$.
• Use the Bogomolov semistability condition on $V$ to derive the improved slope bound $\delta_B/\lambda_B \leq 7 + \frac{6}{g}$, which is stronger than the generic bound.
• Analyze the geometry of the Maroni locus in $\overline{\mathfrak{T}}_g$, defined as the closure of trigonal curves not embeddable in $\mathbb{F}_0$ or $\mathbb{F}_1$, and show that maximal slope families lie entirely within it.
• Compute the rational Picard group of $\overline{\mathfrak{T}}_g$ in even genus by relating divisor classes to the boundary and Hodge classes via linear relations.
• Apply intersection theory on the total space $X$ of the fibration, computing contributions from non-root components and ramification to derive the key linear relation between $\lambda|_B$, $\delta|_B$, and boundary divisors.
• Use the difference $\mathfrak{S}_h = (8g+4)\lambda|_B - g\delta|_B$ to construct an effective linear combination $\mathcal{E}_h$ of boundary divisors, leading to the fundamental relation in $\operatorname{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$.

Experimental results

Research questions

• RQ1 What is the exact upper bound for the slope $\delta_B/\lambda_B$ of non-isotrivial trigonal fibrations over a base curve $B$?
• RQ2 How does the Bogomolov semistability of the canonical rank two vector bundle $V$ associated to a trigonal family constrain the slope of the fibration?
• RQ3 What is the geometric significance of the Maroni locus in $\overline{\mathfrak{T}}_g$, and how is it related to families achieving maximal slope?
• RQ4 How can the rational Picard group of the trigonal locus $\overline{\mathfrak{T}}_g$ be described in terms of boundary and Hodge classes?
• RQ5 What linear relations exist between the Hodge class $\lambda$ and boundary divisor classes in $\operatorname{Pic}_{\mathbb{Q}}\overline{\mathfrak{T}}_g$?

Key findings

• The exact upper bound for the slope of any non-isotrivial trigonal fibration is $\frac{36(g+1)}{5g+1}$, and this bound is sharp.
• Equality in the slope bound is achieved if and only if all fibers are irreducible, $X$ is a triple cover of a ruled surface $Y$ over $B$, and the divisor class $\eta$ on $X$ is numerically zero.
• For Bogomolov-semistable associated vector bundles $V$, the slope satisfies the stronger inequality $\delta_B/\lambda_B \leq 7 + \frac{6}{g}$, which is not the global maximum due to the existence of the Maroni locus.
• The Maroni locus in $\overline{\mathfrak{T}}_g$ consists of trigonal curves that do not embed in $\mathbb{F}_0$ or $\mathbb{F}_1$, and all families achieving the maximal slope $\frac{36(g+1)}{5g+1}$ are contained within this locus.
• The rational Picard group of $\overline{\mathfrak{T}}_g$ in even genus is described via a linear relation involving the Hodge class $\lambda$ and boundary divisor classes, with coefficients derived from intersection-theoretic computations on the fibration.
• The paper establishes a fundamental relation in $\operatorname{Pic}_{\mathbb{Q}}\overline{\mathfrak{I}}_g$: $(8g+4)\lambda = g\xi_0 + \sum_{i=1}^{[(g-1)/2]} 2(i+1)(g-i)\xi_i + \sum_{j=1}^{[g/2]} 4j(g-j)\delta_j$, which generalizes the slope bound to the hyperelliptic case.