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 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.
Bibliographical noteFunding Information:
The second author was partially supported by NSERC grant RGPIN-2017-05732.
We thank Sergio Da Silva, Elisa Gorla, Kuei-Nuan Lin, Yi-Huang Shen, Adam Van Tuyl, and Anna Weigandt for helpful conversations. We are also grateful to the anonymous referee for a very careful reading of the paper and for helpful feedback. Part of this work was completed at the Banff International Research Station (BIRS) during the Women in Commutative Algebra workshop in October 2019. We are grateful for the hospitality of the Banff Centre. The second author was partially supported by NSERC grant RGPIN-2017-05732.
- Geometric vertex decomposition
- Gröbner bases
- Gröbner degeneration