Some Remarks on H-stability of syzygy bundle on algebraic surface

TL;DR

This paper investigates the stability properties of syzygy bundles on algebraic surfaces, proving their stability on various classes of surfaces including Hirzebruch, del Pezzo, and Enriques surfaces, with specific results on $(-K_X)$-stability.

Contribution

It establishes the $L$-stability of syzygy bundles on several types of algebraic surfaces, extending known results and providing new stability conditions.

Findings

Proves $L$-stability of $M_L$ on Hirzebruch surfaces

Establishes $L$-stability of $M_L$ on del Pezzo and Enriques surfaces

Shows $(-K_X)$-stability of syzygy bundles on del Pezzo surfaces

Abstract

Let $L$ be a globally generated line bundle over a smooth irreducible complex projective surface $X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. We prove the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $(-K_X)$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is also obtained.