Three invariants of geometrically vertex decomposable ideals

Thái Thành Nguyễn, Jenna Rajchgot and Adam Van Tuyl

Vol. 333 (2024), No. 2, 357–390

DOI: 10.2140/pjm.2024.333.357

Abstract

We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo–Mumford regularity, the multiplicity, and the $a$-invariant. We show that these invariants can be computed recursively using the ideals that appear in the geometric vertex decomposition process.

As an application, we prove that the $a$-invariant of a geometrically vertex decomposable ideal is nonpositive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley–Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the $a$-invariant of a toric ideal of a bipartite graph.

Keywords

geometrically vertex decomposable, Castelnuovo–Mumford regularity, $a$-invariant, multiplicity, Hilbertian, toric ideals of graphs

Mathematical Subject Classification

Primary: 05E40, 13P10, 14M25

Milestones

Received: 21 February 2024

Revised: 8 December 2024

Accepted: 10 December 2024

Published: 28 December 2024