High rank linear syzygies on low rank quadrics

TL;DR

This paper investigates the structure of linear syzygies in polynomial ideals, identifying conditions that produce high-rank syzygies and providing explicit constructions of toric varieties with such properties.

Contribution

It reveals obstructions to Eisenbud's conjecture on syzygies and constructs toric varieties with high-rank minimal linear syzygies, advancing understanding of syzygy behavior.

Findings

Obstructions to Eisenbud's conjecture identified

Explicit constructions of toric varieties with high-rank syzygies

Counterexamples to expected syzygy rank behavior

Abstract

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.