High rank linear syzygies on low rank quadrics
type: Item
instance of: scholarly article
title: High rank linear syzygies on low rank quadrics
zbMATH Open document ID: 1246.14066
author: Hal Schenck
author: Mike Stillman
published in: American Journal of Mathematics
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.0933
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 6029059
zbMATH Keywords: Syzygy
zbMATH Keywords: Koszul cohomology
zbMATH Keywords: toric variety
MaRDI profile type: Publication
OpenAlex ID: W1978758381
arXiv ID: 1012.0933
DOI: 10.1353/AJM.2012.0012
review text: 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 \(2\).
reviewed by: Paola Bonacini
description: scientific article; zbMATH DE number 6029059