Stability of Tautological Bundles on Symmetric Products of Curves

TL;DR

The paper proves stability and semi-stability of tautological bundles on symmetric products of curves under certain slope conditions, extending stability properties from the original bundle to its tautological counterpart.

Contribution

It establishes new stability criteria for tautological bundles on symmetric products of curves based on the slope of the original bundle.

Findings

Tautological bundles are stable if the original bundle's slope is outside [-1, n-1].

Tautological bundles are semi-stable if the original bundle's slope is outside (-1, n-1).

Results apply to smooth projective curves over complex numbers.

Abstract

We prove that, if $C$ is a smooth projective curve over the complex numbers, and $E$ is a stable vector bundle on $C$ whose slope does not lie in the interval $[-1,n-1]$, then the associated tautological bundle $E^{[n]}$ on the symmetric product $C^{(n)}$ is again stable. Also, if $E$ is semi-stable and its slope does not lie in the interval $(-1,n-1)$, then $E^{[n]}$ is semi-stable.