[Paper Review] Slopes of trigonal fibred surfaces and of higher dimensional fibrations

Miguel Ángel Barja, Lidia Stoppino|arXiv (Cornell University)|Nov 20, 2008

Geometric and Algebraic Topology14 references3 citations

TL;DR

This paper establishes a sharp lower bound for the slope of trigonal fibred surfaces of even genus with Maroni invariant zero, proving a conjecture by Harris and Stankova-Frenkel without requiring semistability. It extends the Cornalba-Harris method to higher-dimensional fibrations, yielding new slope inequalities for canonical fibrations and demonstrating the necessity of GIT semistability in the method's assumptions.

ABSTRACT

We give lower bounds for the slope of higher dimensional fibrations over curves under conditions of GIT-semistability of the fibres, using a generalization of a method of Cornalba and Harris. With the same method we establish a sharp lower bound for the slope of trigonal fibrations of even genus and general Maroni invariant; in particular this result proves a conjecture due to Harris and Stankova-Frenkel.

Motivation & Objective

• To establish a sharp lower bound for the slope of trigonal fibred surfaces with even genus and Maroni invariant zero.
• To extend the Cornalba-Harris method to higher-dimensional fibrations under GIT semistability conditions.
• To prove a new slope inequality for fibrations of canonical varieties in dimension ≥3.
• To provide evidence for the necessity of the GIT semistability assumption in the Cornalba-Harris theorem.
• To investigate the influence of gonality on slope bounds in fibred surfaces, advancing toward a conjectural increasing slope with gonality.

Proposed method

• Generalizes the Cornalba-Harris method to higher-dimensional fibrations using GIT semistability of the fibres.
• Applies the method to fibrations where the relative dualizing sheaf induces a generically finite map or embedding.
• Uses the inequality $ L^n \geq n \cdot \frac{d}{h^0(F, \mathcal{L}|_F)} \cdot \deg f_*\mathcal{L} $, where $ d $ is the degree of the image under the linear system.
• Applies the method to canonical fibrations with $ p_g = n+1 $, $ K_F^{n-1} = n+2 $, and log-terminal singularities.
• Employs Hilbert semistability of the canonical embedding and vanishing of higher direct images to apply the main inequality.
• Uses the construction of Tan to realize curves as fibres of semistable fibrations, proving sharpness of the bound.

Experimental results

Research questions

• RQ1 Can the Cornalba-Harris method be extended to produce slope bounds for higher-dimensional fibrations?
• RQ2 What is the sharp lower bound for the slope of trigonal fibrations of even genus with Maroni invariant zero?
• RQ3 Does the Harris-Stankova-Frenkel conjecture on the slope bound $ \frac{5g-6}{g} $ hold without the semistability assumption?
• RQ4 How does the gonality of the general fibre influence the slope of fibred surfaces?
• RQ5 Is the GIT semistability condition necessary for the Cornalba-Harris method to yield effective slope bounds?

Key findings

• The paper proves the Harris-Stankova-Frenkel conjecture: for a relatively minimal trigonal fibration of even genus $ g \geq 6 $ with Maroni invariant zero, the slope satisfies $ s(f) \geq \frac{5g-6}{g} $, even without semistability.
• The bound $ \frac{5g-6}{g} $ is sharp and is achieved by fibrations constructed via Tan's method from curves with Maroni invariant $ c < (g+2)/9 $.
• For higher-dimensional fibrations, the paper establishes the inequality $ K_f^n \geq \frac{n(n+2)}{n+1} \deg f_*\omega_f $ for $ \mathbb{Q} $-factorial fibrations with canonical fibres of dimension $ n-1 $, $ p_g = n+1 $, and $ K_F^{n-1} = n+2 $.
• The method yields the first known slope inequality for fibrations in dimension greater than 3.
• The result confirms that the GIT semistability assumption is essential: the method fails for higher Maroni invariant cases due to Chow instability.
• The paper shows that the bound $ \frac{5g-6}{g} $ improves upon Konno’s earlier bound $ \frac{14(g-1)}{3g+1} $ for genus 6, and is sharp for genus 6 with a $ g^2_5 $.