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This study explores the relationship between vector bundles on curves and their Jacobians through resonance properties.

June 17, 2026Open Access

On the Resonance Varieties of Vector Bundles on Curves

Key Points

This study explores the relationship between vector bundles on curves and their Jacobians through resonance properties.
Analyzed vector bundles E on a smooth curve X and F on another curve Y of the same rank and degree.
Focused on curves of genus g and examined conditions of positivity and birational resonance in degree c.
If vector bundles E and F have birational resonance in degree c≥g+1, X and Y exhibit isomorphic Jacobians.
Investigated resonance nature on a singular curve with arithmetic genus 1, expanding the understanding of vector bundles.

Abstract

We use the resonance (Papadima, Sociu, Aprodu and others) to study vector bundles E on a smooth curve X of genus g. Our key observation is that if we know it for a specific integer c, then we get strong information on X and E. Fix an integer c≥g+1. Take genus g curves X and Y and vector bundles E on X and F on Y with the same ranks and degrees. Our main result is that if E and F are sufficiently positive and they have birational resonance in degree c, then X and Y have isomorphic Jacobians. We also study the resonance for a singular curve with arithmetic genus 1.

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On the Resonance Varieties of Vector Bundles on Curves

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Abstract

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