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Moment-angle complexes from simplicial posets

100%
Lü Z., Panov T.
Open Mathematics
|
2011
|
tom 9
|
nr 4
715-730
EN
We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.
2
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Derived category of toric varieties with small Picard number

100%
Costa L., Miró-Roig R.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1280-1291
EN
This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D b(X), where X is the blow up of ℙn−r ×ℙr along a multilinear subspace ℙn−r−1×ℙr−1 of codimension 2 of ℙn−r ×ℙr. As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.
3
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Quotients of an affine variety by an action of a torus

76%
Chuvashova O., Pechenkin N.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1863-1880
EN
Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/C T and the toric Hilbert scheme H. We introduce a notion of the main component H 0 of H, which parameterizes general T-orbit closures in X and their flat limits. The main component U 0 of the universal family U over H is a preimage of H 0. We define an analogue of a universal family WX over the main component of X/C T. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U 0 to W X. The variety W X also provides a geometric realization of the Altmann-Hausen family. In particular, the notion of W X allows us to provide an explicit description of the fan of the Altmann-Hausen family in the toric case.
4
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Three-dimensional terminal toric flips

64%
Fujino O., Sato H., Takano Y., Uehara H.
Open Mathematics
|
2009
|
tom 7
|
nr 1
46-53
EN
We describe three-dimensional terminal toric flips. We obtain the complete local description of three-dimensional terminal toric flips.
5
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On orbits of the automorphism group on an affine toric variety

64%
Arzhantsev I., Bazhov I.
Open Mathematics
|
2013
|
tom 11
|
nr 10
1713-1724
EN
Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.
6
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On the graph labellings arising from phylogenetics

52%
Buczyńska W., Buczyński J., Kubjas K., Michałek M.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1577-1592
EN
We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.
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