High rank linear syzygies on low rank quadrics

Abstract

We study the linear syzygies of a homogeneous ideal $I \subseteq S = {\rm Sym}_{{\bf k}}(V)$, focussing on the graded betti numbers $$ b_{i,i+1} = {\rm dim}_{{\bf k}} {\rm Tor}_i (S/I,{\bf k})_{i+1}. $$ For a variety $X$ and divisor $D$ with $V = H^0(D)$, what conditions on $D$ ensure that $b_{i,i+1} \ne 0$? Eisenbud has shown that a decomposition $D \sim A +B$ 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 $\le 4$, 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.