Genuinely ramified maps and stability of pulled-back parabolic bundles

TL;DR

This paper investigates how genuinely ramified maps between smooth projective curves affect the stability of pulled-back parabolic bundles, establishing conditions under which stability is preserved.

Contribution

It proves that under certain conditions, the pullback of a stable parabolic bundle remains stable, extending understanding of stability under ramified covers.

Findings

Pullback of stable parabolic bundles remains stable under genuinely ramified maps.

Stability preservation holds when the branch data is tame or the map is Galois.

Conditions on branch data ensure linear disjointness for stability results.

Abstract

Let $f : X \rightarrow Y$ be a genuinely ramified map between irreducible smooth projective curves defined over an algebraically closed field. Let $P$ be a branch data on $Y$ such that $P(y)$ and $B_f(y)$ where $B_f$ is branch data for $f$ are linearly disjoint for every $y \in, Y$. Further assume that either $P$ is tame or $f$ is Galois. Then the pullback, by $f$, of any stable parabolic bundle on $Y$ with respect to $P$ is actually a stable parabolic bundle on $X$ with respect to $f^*P$.