A simple observation concerning the vector of Riemann constants and non-special divisors of generalized Fermat curves

A closed Riemann surface \( S \) is called a generalized Fermat curve of type \( (k,n) \), where \( k,n \geq 2 \) are integers such that \( (k-1)(n-1) > 2 \), if it admits a group \( H \cong \mathbb{Z}_{k}^{n} \) of conformal automorphisms such that the quotient orbifold \( S/H \) has genus zero and has exactly \( n+1 \) conical points, each of them of order \( k \). If an element of \( H \), of order \( k \), has fixed points, then it has exactly \( k^{\,n-1} \) fixed points, say \( q_{1}, \ldots, q_{k^{\,n-1}} \in S \). To each \( q_{j} \) we associate its vector of Riemann constants \( -2{\mathcal K}_{q_{j}} \in JS \), where \( JS \) is the Jacobian variety of \( S \). Our first observation is that \( {\mathcal K}_{q_{1}} + \cdots + {\mathcal K}_{q_{k^{\,n-1}}} \) is an order two torsion point in \( JS \). Let \( D \) be an effective divisor of degree \( g_{k,n} \), the genus of \( S \). We observe that \( D \) cannot be \( H \)-invariant. In the case that \( D \) is supported on the fixed points of the non-trivial elements of \( H \), then we obtain algebraic conditions, necessary and sufficient, for \( D \) to be non-special.