High rank linear syzygies on low rank quadrics (Q2881362)

The authors study the linear syzygies of a homogeneous ideal \(I\subset S=\mathrm{Sym}_{k}(V)\), with \(k\) algebraically closed field of characteristic \(0\), focusing on the graded Betti numbers: NEWLINE\[NEWLINE b_{i,i+1}=\dim_{k}\mathrm{Tor}_i(S/I,k)_{i+1}. NEWLINE\]NEWLINE Given a variety \(X\) and a divisor \(D\), if \(V=H^0(D)\), which conditions on \(D\) ensure that \(b_{i,i+1}\neq 0\)? 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 \(I_X\); he conjectured that, if \(I_2\) is generated by quadrics of rank \(\leq 4\), then the last non vanishing of \(b_{i,i+1}\) is a consequence of such equations. The authors provide an infinite class of counterexamples to this conjecture, by using ideals that are toric specializations of Rees algebra of Koszul cycles, and prove a variant of the conjecture, with additional hypothesis such as requiring a top linear syzygy of low rank. The authors also give an explicit construction of toric varieties with minimal linear syzygies of arbitrary rank, providing one answer to a question posed by Eisenbud and Koh. In the last section of the paper the authors explore geometric reasons for the failing of the given conjecture, focusing on the case of curves and toric surfaces, under the condition that the top linear syzygies are in degree \(2\).