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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.

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)
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