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Let X be an algebraic curve and c be the Clifford index of X. When c is small (e.g c=1), what is the classification of the line bundle who computes c?

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I know this is an old question (ok, two weeks, whatever) but my guess as to why you haven't gotten any responses is that your question is terribly vague. What are you looking for exactly? A means of constructing a line bundle achieving the bound? Also, you've given no motivation whatsoever, merely asked a hard-to-follow question. I'll gladly attempt to answer the question, once I'm more certain of what it is. –  Charles Siegel Jan 7 '10 at 14:36
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3 Answers 3

Curves of Clifford index 1 (and some other small values) are classified. One should look at old papers written by Gerriet Martens

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when c= 0 Clifford's them includes the fact that any divisor with Clifford index 0 is a multiple of the hyperelliptic fiber, ie: a sum of fibers of the hyperelliptic map. If c=1 then the curve is either trigonal or a plane quintic- I believe that it is an exercise in A-C-G-H. Kind of a folk lore result. I have not heard of anyone explicating all the possible cases when c=2.

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I believe it has been known for a long time that Cliff = 2 occurs only for curves with either a g(1,4) or g(2,6), i.e. roughly 4 - fold covers of P^1 or plane sextics.

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