On curves of odd gonality
Let \(C\) be a smooth irreducible projective curve of odd gonality, that is the minimal number of sheets of a covering over \(\mathbb{P}^1\) is odd and let \(W^r_d (C)\) be the variety of special divisors of degree \(d\) and projective dimension at least \(r\) on \(C\). -- \textit{M. Coppens} and \textit{G. Martens} [Compos. Math. 78, No. 2, 193-212 (1991; Zbl 0741.14035)] gave a refinement of Clifford's theorem for curves with odd gonality, namely they show that for any effective divisor \(D\) on \(C\) with \(\deg D<g\), the genus of \(C\), \(\dim D\leq 1/3 \deg D\). -- \textit{M. Coppens}, \textit{C. Keem} and \textit{G. Martens} [Manuscr. Math. 77, No. 2/3, 237-264 (1992; Zbl 0786.14016)] showed that \(\dim W^r_d (C)\leq d-3r\), again for \(C\) with odd gonality. In the present paper the author improves the above results by giving a complete description of the curves \(C\) and the effective divisors \(D\) for which equalities occur.
scientific article; zbMATH DE number 938578