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Abstract The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts.We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme.We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.
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Combinatorial aspects of virtually Cohen-Macaulay sheaves
There is an abundance of deep literature on the use of free resolutions to study modules and vector bundle resolutions to study coherent sheaves. When studying a module over the Cox ring of a smooth projective toric variety X, each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which are amenable to algebraic and combinatorial study and also capture desirable geometric information. In this extended abstract, we continue this program in the combinatorially-rich Stanley–Reisner setting. In particular, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings. After augmenting the simplicial complexes associated to these Stanley–Reisner rings with a coloring that reflects the product structure on X, our primary tool is Reisner’s criterion, whose conclusion we interpret in the virtual setting. We also provide two constructions of short virtual resolutions for use beyond the Stanley–Reisner case.
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- Award ID(s):
- 2001101
- PAR ID:
- 10340047
- Editor(s):
- Bauer, Tomer; Liu, Gaku
- Date Published:
- Journal Name:
- Seminaire lotharingien de combinatoire
- Volume:
- 85B
- ISSN:
- 1286-4889
- Page Range / eLocation ID:
- Art. 263
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/tlm3.12036
