The primitive length of a general k-gonal curve

Linear Series on 4 - Gonal Curves

“…Since it is simple again it cannot be of type 1. Thus, according to [CKM2,4.2] we conclude that its Clifford index is at least g−2 2 , and so 2d ≥ g + 4r − 2 + 2i ≥ g + 4r − 2.…”

“…For k ≥ 3 and r = 1 there is the following existence theorem ( [CKM2]): On the general k -gonal curve C (k ≥ 3) there are non -trivial pencils g 1 d of degree d > k if and only if g+1 2 < d < g. Note that this bound is slightly different from that of Proposition 1.16 for r = 1. If k ≥ 3 and r = 2, however, B. Segre ( [S]; see also [AC]) proved that the inequalities in Proposition 1.16 are also sufficient for the existence of type 3 -nets on the general k -gonal curve.…”

“…Proposition 1.15) and deg F = u(2) − 4 = g−1 2 which is minimal by Proposition 1.16). For r = 3 we let g 3 u(3) = g 1 4 + F where |F | denotes a (complete and) base point free pencil of degree g+3 2 ( [CKM2,2.2.3]). Since |F | is not composed of the g 1 4 this web g 3 u(3) is simple (e. g., cf.…”

“…To see this observe that any type 1 -series rg 1 k on C is primitive. In fact, 2 ≤ h 1 rg 1 k = g − rk + r = g − (k − 1)r, i. e. r ≤ g−2 k−1 , and so, by [CKM2,1.2] the rg 1 k is primitive. Consequently, if some type 1 -series rg 1 k were of type 2 we would have…”

On the existence of special stable spanned vector bundles on projective curves

“…Theorem 0.1 is easy to use both for curves with general moduli and for curves for which the classical theory of special divisor is known. The interested reader may see the case of plane curves in [5]. The proof of 0.1 uses a construction (see 1.1) made in [2] for the same purpose.…”