On the stability of pulled back parabolic vector bundles

Abstract

Take an irreducible smooth projective curve $X$ defined over an algebraically closed field of characteristic zero, and fix finitely many distinct point $D=\{x_1,\cdots ,x_n\}$ of it; for each point $x\in D$ fix a positive integer $N_x$. Take a nonconstant map $f:Y⟶X$ from an irreducible smooth projective curve. We construct a natural subbundle $\mathcal{F}\subset f_{\ast }{\mathcal{O}}_Y$ using $(D,\{N_x{\}}_{x\in D})$. Let $E_{\ast }$ be a stable parabolic vector bundle whose parabolic weights at each $x\in D$ are integral multiples of $\frac{1}{N_x}$. We prove that the pullback $f^{\ast }{E}_{\ast }$ is also parabolic stable, if $\mathrm{rank}(\mathcal{F})=1$.

Final version