Let \(d\ge 2\) be a positive integer, K an algebraically closed field of characteristic not dividing d, \(n\ge d+1\) a positive integer prime to d, \(f(x)\in K[x]\) a degree n monic polynomial without repeated roots, \(C_{f,d}: y^d=f(x)\) the corresponding smooth plane affine curve over K, and \(\mathcal {C}_{f,d}\) a smooth projective model of \(C_{f,d}\) . Let \(J(\mathcal {C}_{f,d})\) be the Jacobian of \(\mathcal {C}_{f,d} \) . We identify \(\mathcal {C}_{f,d}\) with the image of its canonical embedding into \(J(\mathcal {C}_{f,d})\) (such that the infinite point of \(\mathcal {C}_{f,d}\) goes to the zero of the group law on \(J(\mathcal {C}_{f,d})\) ). Earlier the second named author proved that if \(d=2\) and \(n=2g+1 \ge 5\) , then the genus g hyperelliptic curve \(\mathcal {C}_{f,2}\) contains no torsion points of orders lying between 3 and \(n-1=2g\) . In the present paper we generalize this result to the case of arbitrary d. Namely, we prove that if P is a torsion point of order \(m>1\) on \(\mathcal {C}_{f,d}\) , then either \(m=d\) or \(m\ge n\) . We also describe all curves \(\mathcal {C}_{f,d}\) having a torsion point of order n.
