Stability of tautological bundles on symmetric products of curves

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 $[-1, n-1]$, then $E^{[n]}$ is semi-stable.

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Received 5 March 2019

Accepted 2 July 2019

Published 17 February 2021