17,915 research outputs found
Binomial Difference Ideal and Toric Difference Variety
In this paper, the concepts of binomial difference ideals and toric
difference varieties are defined and their properties are proved. Two canonical
representations for Laurent binomial difference ideals are given using the
reduced Groebner basis of Z[x]-lattices and regular and coherent difference
ascending chains, respectively. Criteria for a Laurent binomial difference
ideal to be reflexive, prime, well-mixed, perfect, and toric are given in terms
of their support lattices which are Z[x]-lattices. The reflexive, well-mixed,
and perfect closures of a Laurent binomial difference ideal are shown to be
binomial. Four equivalent definitions for toric difference varieties are
presented. Finally, algorithms are given to check whether a given Laurent
binomial difference ideal I is reflexive, prime, well-mixed, perfect, or toric,
and in the negative case, to compute the reflexive, well-mixed, and perfect
closures of I. An algorithm is given to decompose a finitely generated perfect
binomial difference ideal as the intersection of reflexive prime binomial
difference ideals.Comment: 72 page
Characteristic classes of singular toric varieties
In this paper we compute the motivic Chern classes and homology Hirzebruch
characteristic classes of (possibly singular) toric varieties, which in the
complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As
special cases, we obtain new (or recover well-known) formulae for the
Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties,
as well as for the Thom-Milnor L-classes of simplicial projective toric
varieties. We present two different perspectives for the computation of these
characteristic classes of toric varieties. First, we take advantage of the
torus-orbit decomposition and the motivic properties of the motivic Chern and
resp. homology Hirzebruch classes to express the latter in terms of dualizing
sheaves and resp. the (dual) Todd classes of closures of orbits. This method
even applies to torus-invariant subspaces of a given toric variety. The
obtained formula is then applied to weighted lattice point counting in lattice
polytopes and their subcomplexes, yielding generalized Pick-type formulae.
Secondly, in the case of simplicial toric varieties, we compute our
characteristic classes by using the Lefschetz-Riemann-Roch theorem of
Edidin-Graham in the context of the geometric quotient description of such
varieties. In this setting, we define mock Hirzebruch classes of simplicial
toric varieties and investigate the difference between the (actual) homology
Hirzebruch class and the mock Hirzebruch class. We show that this difference is
localized on the singular locus, and we obtain a formula for it in which the
contribution of each singular cone is identified explicitly. Finally, the two
methods of computing characteristic classes are combined for proving several
characteristic class formulae originally obtained by Cappell and Shaneson in
the early 1990s.Comment: v2: new references added; many results hold now in greater
generality, e.g. for closed algebraic toric invariant subspaces of toric
varieties which have only DuBois singularities by work of Ishida; motivic
Chern classes are also computed; new examples are worked out in detai
Permutation actions on equivariant cohomology
This survey paper describes two geometric representations of the permutation
group using the tools of toric topology. These actions are extremely useful for
computational problems in Schubert calculus. The (torus) equivariant cohomology
of the flag variety is constructed using the combinatorial description of
Goresky-Kottwitz-MacPherson, discussed in detail. Two permutation
representations on equivariant and ordinary cohomology are identified in terms
of irreducible representations of the permutation group. We show how to use the
permutation actions to construct divided difference operators and to give
formulas for some localizations of certain equivariant classes.
This paper includes several new results, in particular a new proof of the
Chevalley-Monk formula and a proof that one of the natural permutation
representations on the equivariant cohomology of the flag variety is the
regular representation. Many examples, exercises, and open questions are
provided.Comment: 24 page
Extensions of toric line bundles
For any two nef line bundles F and G on a toric variety X represented by
lattice polyhedra P respectively Q, we present the universal equivariant
extension of G by F under use of the connected components of the set theoretic
difference of Q and P
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