For smooth abelian Galois covers, a relatively Ulrich bundle exists exactly when each branch divisor is a sum of d_i-fold tensor products of global sections; so every smooth abelian cover of P^n has an Ulrich bundle.
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On the triviality of direct image of coherent sheaves
For smooth abelian Galois covers, a relatively Ulrich bundle exists exactly when each branch divisor is a sum of d_i-fold tensor products of global sections; so every smooth abelian cover of P^n has an Ulrich bundle.