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Toric P-difference varieties. (English) Zbl 1453.12007
Summary: In this paper, we introduce the concept of P-difference varieties and study the properties of toric P-difference varieties. Toric P-difference varieties are analogues of toric varieties in difference algebraic geometry. The category of affine toric P-difference varieties with toric morphisms is shown to be antiequivalent to the category of affine $$P[x]$$-semimodules with $$P[x]$$-semimodule morphisms. Moreover, there is a one-to-one correspondence between the irreducible invariant P-difference subvarieties of an afne toric P-difference variety and the faces of the corresponding affine $$P[x]$$-semimodule. We also define abstract toric P-difference varieties by gluing affine toric P-difference varieties. The irreducible invariant P-difference subvariety-face correspondence is generalized to abstract toric P-difference varieties. By virtue of this correspondence, a divisor theory for abstract toric P-difference varieties is developed.
MSC:
 12H10 Difference algebra 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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