A generalization of Bring’s curve in any characteristic

Abstract A natural generalization of Bring’s curve, valid over any field 𝕂 of characteristic zero or characteristic p ≥ 7 p\geq 7 , is the algebraic variety 𝑉 of PG ⁡ ( m − 1 , K ) \operatorname{PG}(m-1,\mathbb{K}) , m ≥ 5 m\geq 5 , which is the complete intersection of the projective algebraic hypersurfaces of the homogeneous equations x 1 k + ⋯ + x m k = 0 x_{1}^{k}+\cdots+x_{m}^{k}=0 with 1 ≤ k ≤ m − 2 1\leq k\leq m-2 . In positive characteristic, we also assume m ≤ p − 1 m\leq p-1 . Up to a change of coordinates in PG ⁡ ( m − 1 , K ) \operatorname{PG}(m-1,\mathbb{K}) , we show that 𝑉 is a projective, absolutely irreducible, non-singular curve of PG ⁡ ( m − 2 , K ) \operatorname{PG}(m-2,\mathbb{K}) . We show that if the automorphism group Aut ⁡ ( V ) \operatorname{Aut}(V) is tame (in particular in characteristic zero), then