We first prove the following: Let C be a smooth bicanonically embedded curve, then Sec^j(C) has determinantal equations iff j < Cliff(C). Examining the proof leads to a generalization of the Clifford index to an arbitrary (very ample) line bundle L. This leads to a similar theorem stating when C and certain secant varieties embedded in L \otimes L have determinatal equations. If time permits the generalizations to L_1 \otimes L_2 and the proof of the Eisenbud-Koh-Stillman conjecture will be discussed.

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