Abstract
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.
Highlights
Determinantal ideals and their generalisations have been explored extensively both in the context of commutative algebra and in the study of Schubert varieties in flag varieties
◦ each two-sided mixed ladder determinantal ideal is a type A Kazhdan–Lusztig ideal; ◦ each ideal generated by the k × k minors of a generic symmetric matrix is the defining ideal of an open patch of a Schubert variety in a Lagrangian Grassmannian; ◦ each defining ideal of a variety of complexes is a type A Kazhdan–Lusztig ideal, up to some extra indeterminate generators
We establish an explicit connection between geometric vertex decomposition and liaison, and we study implications of this connection
Summary
Determinantal ideals and their generalisations have been explored extensively both in the context of commutative algebra and in the study of Schubert varieties in flag varieties. Yong introduced geometric vertex decomposition, a degeneration technique, and used it to study the Gröbner geometry of Schubert determinantal ideals for vexillary permutations. The first goal of this paper is to show that it is no coincidence that geometric vertex decomposition and liaison can be used to obtain similar results for similar classes of ideals. If the Lex-initial ideal of I is the Stanley– Reisner ideal of a vertex decomposable simplicial complex and the vertex decomposition is compatible with the order of the variables, I is glicci From this corollary, one can quickly deduce that certain well-known classes of varieties are glicci. Nagel and Römer showed, more generally, that the Stanley–Reisner ideal of a weakly vertex decomposable simplicial complex is glicci [34, Theorem 3.3]. Throughout the paper, we let κ be a field, which can be chosen arbitrarily except in Sections 4, 5 and 7, where we require that κ be infinite
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