Uniform Trace-free Bundles of Triple Covers over Projective Planes

TL;DR

This paper proves that for certain triple covers over the projective plane with specific branch divisor degrees, the associated trace-free bundles are uniform, and provides a complete classification of these bundles.

Contribution

It establishes conditions under which the trace-free bundle of a normal triple cover over 2 is uniform and offers a full classification of such bundles.

Findings

Trace-free bundles are uniform when the branch divisor degree is at most 18 and not 16.

Complete classification of these trace-free bundles.

Identifies specific degree conditions for uniformity.

Abstract

Studying coverings over algebraic varieties is an effective method in algebraic geometry. By combining the technique of triple cover from Miranda and Tan, we proved that if the degree of the branch divisor of a normal triple cover over $\P^2$ is no more than $18$ and not equal to $16$, then the trace-free bundle of the triple cover is uniform. Moreover, we totally classified all these trace-free bundles.