Chow rings of toric varieties defined by atomic lattices

By Eva Maria Feichtner, Sergey Yuzvinsky. 2003

184 citations recorded by the source corpus.

We study a graded algebra $D=D(\mathcal{L},\mathcal{G})$ over ℤ defined by a finite lattice ℒ and a subset $\mathcal{G}$ in ℒ, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice ℒ, as the Chow ring of a smooth toric variety that we construct from ℒ and $\mathcal{G}$. We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Gröbner basis of the relation ideal of D and a monomial basis of D.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a2944bb4119c36dc4597ff469d39d42e8cb965b5

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Citations in the corpus

COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS (2017, background): “Feichtner and Yuzvinsky introduced the Chow ring of M in [18].”
Nested complexes and their polyhedral realizations (2005, methodology): “An even more general setup developed by Feichtner - Yuzvinsky in [6] associates a nested complex and a simplicial fan to an arbitrary finite atomic lattice.”
A semi-small decomposition of the Chow ring of a matroid (2020, methodology): “The Chow ring of M, introduced by Feichtner and Yuzvinsky in [17], is the quotient algebra1 CH(M) := SM/(IM + JM), where IM is the ideal generated by the linear forms ∑ i1∈F xF − ∑ i2∈F xF , for every pair of distinct elements i1 and i2 of E,”
Simplicial generation of Chow rings of matroids (2019, unknown): “The ring A • ( M ) was ﬁrst studied in [FY04] under a slightly different presentation, which for clarity is denoted A • FY ( M ) and is given as .”
Incidence combinatorics of resolutions (2003, background): “Moreover, we give a short account of the study of a graded algebra associated with any finite lattice in [10], where our combinatorial generalization of originally geometric notions leads to the construction of an, at first sight, unrelated geometric counterpart for wonderful models of hyperplane ar”