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Under which circumstances is the Clifford index of a curve computed by pencils?

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Almost always. The Clifford dimension of the curve is the smallest $r$ such that the Clifford index is given by a $g^r_d$. Curves of Clifford dimension $>1$ are rather rare. The curves of Clifford dimension 2 are exactly the smooth plane curves of degree $\geq 5$. Curves of Clifford dimension 3 are also known; in general, there are nice conjectures of Eisenbud, Lange, Martens and Schreyer, see here.

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