The primitive length of a general \(k\)-gonal curve
For a (smooth, irreducible projective) curve \(C\) of genus \(g \geq 4\), its Clifford sequence is defined as the set of Clifford indices \((d - 2r)\) for all nontrivial primitive linear series \(g^r_d\) (i.e. \(g^r_d\) is complete, and both it and its dual \(|K_C - g^r_d |\) verify that they are base-point free and are not the trivial \(g^0_0)\). The primitive length of \(C\) is defined to be the cardinality of the Clifford sequence. In Manuscr. Math. 77, No. 2/3, 237-264 (1992; Zbl 0786.14016), the authors began a detailed study of primitive linear series and Clifford lengths of curves. In the paper under review the authors continue this study, concentrating now on the general \(k\)-gonal curve and the Brill-Noether theory. More precisely, let \(3 < k < [(g + 3)/2]\) (the extreme cases where already studied in their previous paper), \(n \leq g\) and let \(\rho^1_n (g) = 2(n - 1) - g\) be the Brill-Noether number. Then the authors prove that there exist components of \(W^1_n\) different from \(W^1_k + W^0_{n - k}\), and that their dimension is given by \(\rho^1_n (g)\) (the problem of the uniqueness of such a component if \(\rho^1_n (g) > 0\) remains open). They also give some evidence for the following conjecture: Any number in the Clifford sequence that is smaller than \([(g - 1)/2]\) is the Clifford index of a multiple of the \(g^1_k\).
scientific article; zbMATH DE number 639722
scientific article