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On a result of Farkas. (English) Zbl 0718.14020

In this short note the authors study the Clifford index c of an algebraic curve C of genus \(g\geq 4\), which improves previous work of H. H. Martens [J. Reine Angew. Math. 233, 89-100 (1968; Zbl 0221.14004)] and H. M. Farkas [ibid. 391, 213-220 (1988; Zbl 0651.14018)]. The Clifford index c is the minimum of the nonnegative integers d-2r, where d is the degree and r the dimension of a linear series L of C with \(r\geq 1\), \(g-1-d+r\geq 1\); such an L is said to compute c in case \(c=d-2r\). Main result:
If L computes c, \(d\leq g-1\) and \(r\geq 3\), then either L is birationally very ample, or C is hyperelliptic, or C is bielliptic.
Using this theorem the authors deduce the following two inequalities relating the genus of C and the degree of an L computing c, for \(c\geq 3:\) \((a)\quad If\) \(3(c+2)/2<d<2(c+2)\), then \(g\leq 2(c+2)\). \((b)\quad If\) \(g\geq 3(c+1)\) and \(d\leq g-1\), then \(d\leq 3(c+2)/2\). From these, they improve Martens’ result to: If \(c\geq 3\) and L computes c and has degree \(d\leq g-1\), then \(d\leq 3c\), and Farkas’ one to: If \(g>3(c+1)\), then there is an L computing c with degree \(d\) such that \(3(c+2)/2<d\leq g-1\) if and only if C is hyperelliptic or bielliptic.
The paper is very concisely written and uses the standard methods of complex algebraic curves (the Riemann-Roch theorem, Castelnuovo’s inequality, etc.).
Reviewer: J.M.Ruiz (Madrid)

MSC:

14H45 Special algebraic curves and curves of low genus
14C20 Divisors, linear systems, invertible sheaves
30F99 Riemann surfaces
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