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



When is the Clifford index of a curve computed by pencils?
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](http://archive.numdam.org/ARCHIVE/CM/CM_1989__72_2/CM_1989__72_2_173_0/CM_1989__72_2_173_0.pdf).


