Abstract: We study the linear syzygies of a homogeneous ideal $I$ in a polynomial ring $S = k[x_0..x_n]$, focussing on the graded betti numbers \[ b_i = {\textrm{dim}}_k {\textrm{Tor}}_i(S/I, k)_{i+1}. \] For any projective variety $X$ in $P^n$ and divisor $D$, what conditions on $D$ ensure that $b_i$ is nonzero? Eisenbud has shown that a decomposition $D=A+B$ such that $A$ and $B$ have at least two sections give rise to determinantal equations (and corresponding syzygies) in $I_X$ and conjectured that if the quadratic component of $I$ is generated by quadrics of rank at most four, then the last nonvanishing $b_i$ is a consequence of such a decomposition. We describe obstructions to the conjecture and prove a variant. The obstructions arise from toric specializations of the Rees algebra of Koszul cycles, and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank. This leads to a number of interesting open questions. (joint work with M. Stillman).