Let C be a complete non-singular curve over ℂ of genus g. We denote by Wdr(C) the subscheme of the Picard variety Picd(C) whose support is the locus of complete linear series of degree d and dimension at least r. In case d > g + r - 1, Wdr (C) = Picd(C) and if d = g + r - 1, Wdr (C) has dimension d. Therefore the dimension of Wdr (C) is independent of C in the range d ≥ g + r - 1. If d ≤ g + r - 2, one knows that dimWdr (C) ≥ p(d, g, r):= g - (r + 1) (g - d + r) for any curve C and is equal to p(d, g, r) for general curve C (see Kleiman-Laksov  and Griffiths-Harris ). But the dimension of Wdr(C) might be greater than p(d, g, r) for some special curve C. Moreover, for curves C with dim Wdr (C) > p(d, g, r), C must be of some special type of curves. The first important result along this line is the following well-known theorem of H. Martens which has been extended by a theorem of D. Mumford.
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