Séminaire Lotharingien de Combinatoire, 85B.13 (2021), 12 pp.
Patricia Klein and Jenna Rajchgot
Geometric Vertex Decomposition and Liaison
Abstract.
Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. 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. We also use the structure of Knutson, Miller, and Yong's geometric vertex decomposition to provide a streamlined implementation of Gorla, Nagel, and Migliore's liaison-theoretic approach to establishing Gröbner bases.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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