Geometry of linearly stable coherent systems over curves

2025/09/14 by Abel Castorena, Castorena, Abel, George H. Hitching +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2509.11244

Abstract

Let E be a vector bundle over a smooth curve C, and V a generating space of sections of E. We characterise Mumford linear stability of the associated projective model of ℙ E^\vee in ℙ V^\vee in terms of geometric and cohomological properties of the coherent system (E, V), and give some applications. We show that any ℙ^r-1-bundle over C has a linearly stable model in ℙ^n-1 for any n ≥ r+2. Furthermore; linear stability of (E, V) is a necessary condition for stability of the kernel bundle M_{E, V} of (E, V), which is predicted by Butler's conjecture for general C and (E, V). We give new examples showing that it is not in general sufficient; in particular, a general bundle E of large degree fits into a linearly stable coherent system (E, V) with nonsemistable kernel bundle. Finally, we use these ideas to show the stability of M_{E, V} for certain (E, V) of type (r, d, r+2) where E is not necessarily stable.

Citations

• Linear stability and rank two Clifford indices of algebraic curves with applications
• Linear stability of coherent systems and applications to Butler's conjecture
• Secant loci of scrolls over curves
• Higher rank Brill-Noether theory and coherent systems: Open questions
• On generated coherent systems and a conjecture of D. C. Butler
• On Chow Stability for algebraic curves
• Clifford indices for vector bundles on curves
• On coherent systems of type (n,d,n+1) on Petri curves
• Maps between moduli spaces of vector bundles and the base locus of the theta divisor
• Birational maps of moduli of Brill-Noether pairs
• Moduli of Brill-Noether pairs on algebraic curves

Related

• Linear stability of coherent systems and applications to Butler's conjecture
• On α-stability and linear stability of generated coherent systems
• On the geometry of moduli spaces of coherent systems on algebraic curves
• Existence of coherent systems
• Moduli spaces of coherent systems of small slope on algebraic curves