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Cohomology of wheels on toric varieties. (English) Zbl 1337.14042
The article studies cohomology groups of four term complexes \(L\rightarrow\bigoplus_{j=1}^{m} L_{j,j+1}\rightarrow\bigoplus_{j=1}^{m} L_{j}\rightarrow L\) for some \(m\geq 2\), where \(L\), \(L_{j}\), and \(L_{j,j+1}\) are invertible sheaves on a normal toric variety. These complexes can be rearranged conveniently into a shape of a bicycle wheel which motivates the title. The work generalizes a result by S. Cautis and T. Logvinenko [J. Reine Angew. Math. 636, 193–236 (2009; Zbl 1245.14016); erratum ibid. 689, 243–244 (2014)] into a broader class of four-term complexes. The authors apply the Cox functor to translate the problem into a commutative algebra problem and use a natural interpretation of the syzygy modules in terms of circuits in the complete graph with \(m\) vertices. The article contains explicit computations on a free-fold performed by hand and with Macaulay2.

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13D02 Syzygies, resolutions, complexes and commutative rings
05E40 Combinatorial aspects of commutative algebra
05C38 Paths and cycles
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Software:
Macaulay2
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