| This paper gives a geometric characterization of the Clifford index of a curve C, in terms of the existence of determinantal equations for C and its secant varieties in the bicanonical embedding. The key idea is the generalization of Shiffer deformations to Shiffer variations supported on an arbitrary effective divisor D. The definition of Shiffer variations and Clifford index is then generalized to an arbitrary very ample line bundle, L, on C. This allows one to give geometric characterizations of the generalized Clifford index in many cases. In particular it allows one to show the existence of determinantal equations for C and Seck( C) for k < Cliff(C, L) in the embedding of C in the projective P (H° L⊗2 ∨ ). We then explain how the results can be generalized to embeddings of C in P (H0 L1⊗L2 ∨ ) where L1 and L 2 are very ample. For line bundles of large degree this proves a conjecture of Eisenbud, Koh, and Stillman relating the existence of a determinantal presentation for Seck(C) to deg( L). |
