Geometric vertex decomposition and liaison

TL;DR

This paper establishes a direct connection between geometric vertex decomposition and liaison theory, showing how these frameworks relate and applying this to prove certain algebraic ideals are glicci.

Contribution

It explicitly links geometric vertex decomposition with liaison theory, demonstrating their equivalence in producing algebraic ideal properties and identifying classes of ideals as glicci.

Findings

Geometric vertex decomposable ideals are linked to ideals of indeterminates via G-biliaisons.

Every G-biliaison of a certain type corresponds to a geometric vertex decomposition.

Several well-known ideals, including Schubert determinantal ideals, are proven to be glicci.

Abstract

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras.