  We study the linear syzygies of a homogeneous ideal I in a polynomial ring S,
focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For
a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure
that b_(i,i+1) is nonzero? Eisenbud has shown that a decomposition D = A + B
such that A and B have at least two sections gives rise to determinantal
equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is
generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1)
is a consequence of such equations. We describe obstructions to this 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 gives one
answer to a question posed by Eisenbud and Koh about specializations of
syzygies.
