Green-Lazarsfeld condition for toric edge ideals of bipartite graphs

[摘要] Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs G whose toric edge ideal I-G is generated by quadrics, showing that every cycle of G of length at least 6 must have a chord. This corresponds to the Green-Lazarsfeld condition N-1. In this paper, we investigate the higher syzygies of I-G and give combinatorial descriptions of the Green-Lazarsfeld conditions N-p of toric edge ideals of bipartite graphs for all p >= 1. In particular, we show that I-G is linearly presented (i.e. satisfies condition N-2) if and only if the bipartite complement of G is a tree of diameter at most 3. We also investigate the regularity of linearly presented toric edge ideals and give criteria for polyomino ideals to satisfy the Green-Lazarsfeld conditions. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15

[关键词] Free resolutions;Toric edge ring;Toric edge ideal;Green-Lazarsfeld conditions;Linearly presented ideals;Regularity