For sextic covers of P1 the possible Tschirnhausen bundles are classified, all constraints arise from algebra multiplication, and every bundle is realized by a cover with a nontrivial proper subcover.
Casnati, Covers of algebraic varieties
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Tschirnhausen bundles of sextic covers of $\mathbb{P}^1$
For sextic covers of P1 the possible Tschirnhausen bundles are classified, all constraints arise from algebra multiplication, and every bundle is realized by a cover with a nontrivial proper subcover.