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Remarks on Clifford indices. (English) Zbl 0651.14018

Let S be a compact Riemann surface of genus \(g\geq 2\). Let r and s be integers with \(0<r\leq s\leq g-1\). The Clifford index of a special divisor on S of (projective) dimension r and degree \(r+s\), is defined as the integer \(s-r\). By the Clifford index of S we mean the smallest of the Clifford indices of special divisors on S. Clifford’s theorem assserts that a surface of Clifford index zero is hyperelliptic, i.e. if there is a special divisor of Clifford index zero then there is a divisor of Clifford index zero and \(r=1.\)
One may ask if a similar result holds for higher values of the Clifford index. In the paper under review it is shown that if the Clifford index t is realized with a divisor of degree d satisfying \(3t+2\leq d\leq g-1\), and if \(g>3(t+1)\), then S carries a meromorphic function of order \(t+2.\) Counterexamples to an unrestricted generalisation are discussed, and a proof of Clifford’s theorem is obtained.
Reviewer: H.Martens

MSC:

14H45 Special algebraic curves and curves of low genus
30F10 Compact Riemann surfaces and uniformization
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