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

H. Torres-López, A. G. Zamora

Published 2020-08-25Version 1

Let $L$ be a globally generated line bundle over a smooth irreducible projective surface$X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. The main theorem proves that the syzygy bundle defined by $nL+D$ is stable for any polarization $H$, where $L$ is any ample , $D$ is an arbitrary divisor and $n$ is a sufficiently large natural number. Taking $n=1$, we obtain the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. Finally, the $-K$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is proved.