On instability of Syzygy Bundles

In this article, we investigate the instability of syzygy bundles corresponding to globally generated vector bundles on smooth irreducible projective surfaces under change of polarization.

💡 Research Summary

The paper investigates how the (semi)stability of syzygy bundles associated to globally generated vector bundles on smooth projective surfaces varies when the polarization is changed. Let X be a smooth irreducible projective surface over the complex numbers and let E be a globally generated vector bundle of rank r with determinant line bundle D = det(E). The syzygy bundle M_E is defined as the kernel of the evaluation map H⁰(X,E)⊗𝒪_X → E, giving the short exact sequence 0 → M_E → H⁰(X,E)⊗𝒪_X → E → 0. Its slope with respect to a polarization H is μ_H(M_E) = –c₁(E)·H/(h⁰(E) – r).

The author reviews earlier work: Butler (1994) proved semistability of M_E on curves under a degree condition, Ein‑Lazarsfeld‑Mustopa (2013) showed that for a very ample line bundle L_d = dH + P on a surface the syzygy bundle M_d becomes H‑stable for d≫0, and subsequent papers extended these results to higher rank bundles but left the behavior under varying polarizations largely open.

The main contributions are two theorems. Theorem 3.3 gives a concrete numerical criterion ensuring that a twist of the syzygy bundle by the line bundle 𝒪_X(–S) has larger slope than the original bundle, i.e. μ_A(M_E(d)⊗𝒪_X(–S)) > μ_A(M_E(d)) for sufficiently large d, provided the inequality
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