The \((g-1)\)-support cover of the canonical locus (Q1082487)

Let \(S\) be a closed Riemann surface of genus \(g\geq 2,\) and let \(S^{(n)}\) be the complex manifold of positive divisors of degree \(n\) on S. The main result of this paper asserts that \(S\) is non-hyperelliptic if and only if the set of points \((D_ 1,D_ 2)\) in \(S^{(g-1)}\times S^{(g-1)}\) such that \(D_ 1+D_ 2\) is canonical, is irreducible. As a corollary one obtains Max Noether's characterization of non- hyperellipticity: the space of holomorphic q-differentials is spanned by products of 1-differentials.