[Paper Review] Moduli of Vector Bundles on Curves in Positive Characteristics

Kirti Joshi, Eugene Z. Xia|ArXiv.org|Oct 4, 1999

Material Science and Thermodynamics4 citations

TL;DR

This paper studies the moduli of semi-stable rank-2 vector bundles on a genus 2 curve over a field of characteristic 2. It shows there is exactly one such bundle (up to tensoring with a line bundle of order 2) whose Frobenius pullback is not semi-stable, and resolves the indeterminacy of the Frobenius map at this point via Higgs bundles, providing a key example in positive characteristic geometry.

ABSTRACT

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.

Motivation & Objective

• To understand the behavior of the Frobenius morphism on the moduli space of rank-2 vector bundles over a genus 2 curve in characteristic 2.
• To analyze the indeterminacy of the Frobenius map at points where the pullback of a semi-stable bundle fails to be semi-stable.
• To resolve this indeterminacy by introducing Higgs bundles into the moduli-theoretic framework.
• To classify semi-stable rank-2 bundles with fixed determinant (a theta characteristic) under the action of tensoring by order-2 line bundles.
• To establish the uniqueness (up to tensoring by order-2 line bundles) of the bundle whose Frobenius pullback is not semi-stable.

Proposed method

• Analyzes the action of the Frobenius morphism on the moduli scheme of rank-2 semi-stable vector bundles on a genus 2 curve over an algebraically closed field of characteristic 2.
• Identifies the locus where the Frobenius pullback of a semi-stable bundle fails to be semi-stable.
• Uses the theory of Higgs bundles to resolve the indeterminacy of the Frobenius map at this singular point.
• Applies the notion of semi-stability and determinant conditions, specifically fixing the determinant to be a theta characteristic.
• Employs deformation-theoretic and moduli-theoretic techniques to classify such bundles up to tensoring by line bundles of order 2.
• Relies on the structure of the moduli space and the behavior of the Frobenius morphism in positive characteristic.

Experimental results

Research questions

• RQ1 What is the structure of the moduli space of rank-2 semi-stable vector bundles on a genus 2 curve in characteristic 2?
• RQ2 Which semi-stable vector bundles have Frobenius pullbacks that are not semi-stable?
• RQ3 How can the indeterminacy of the Frobenius map be resolved at points where the pullback fails to preserve semi-stability?
• RQ4 Is there a unique such bundle (up to tensoring with an order-2 line bundle) with this property?
• RQ5 Can Higgs bundles be used to resolve the rational map induced by Frobenius on the moduli space?

Key findings

• There exists exactly one isomorphism class (up to tensoring with a line bundle of order 2) of semi-stable rank-2 vector bundles with determinant a theta characteristic whose Frobenius pullback is not semi-stable.
• The Frobenius map on the moduli scheme is indeterminate precisely at this unique point.
• The indeterminacy of the Frobenius map at this point is resolved by introducing Higgs bundles into the moduli-theoretic framework.
• The classification of such bundles is complete up to tensoring with line bundles of order 2.
• The result provides a concrete example of the failure of Frobenius to preserve semi-stability and its resolution in positive characteristic geometry.
• The study reveals a deep connection between the geometry of the moduli space and the behavior of the Frobenius morphism in characteristic 2.