A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The result is based on the Cox realization of a toric variety as a quotient space of an open subset of a vector space V by a quasitorus action and on investigation ...

We show how to construct quantum stabilizer and subsystem codes from algebro-geometric toric codes extending the known construction of subsystem codes from cyclic codes and extending the construction of stabilizer codes from toric codes in an earlier work by one of the authors. Since algebrogeometric toric codes are higher dimensional extensions of cyclic codes, we obtain this way a new and ric...

We use a recently proposed formulation of stable holomorphic vector bundles V on elliptically fibered Calabi–Yau n-fold Zn in terms of toric geometry to describe stability conditions on V . Using the toric map f : Wn+1 → (V, Zn) that identifies dual pairs of F-theory/heterotic duality we show how stability can be related to the existence of holomorphic sections of a certain line bundle that is ...

We discuss an experimental approach to open problems in toric geometry: are smooth projective toric varieties (i) arithmetically normal and (ii) defined by degree 2 equations? We discuss the creation of lattice polytopes defining smooth toric varieties as well as algorithms checking properties (i) and (ii) and further potential properties, in Particular a weaker version of (ii) asking for schem...

Toric differential inclusions play a pivotal role in providing rigorous interpretation of the connection between weak reversibility and persistence mass-action systems polynomial dynamical systems. We introduce notion quasi-toric inclusions, which are strongly related to toric inclusions, but have much simpler geometric structure. show that every inclu...

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extends Stanley and Reisner’s face ring for abstract simplicial complexes [20] and Stanley...

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler–Einstein ...

We give examples of proper normal surfaces that are not embeddable into simplicial toric prevarieties nor toric prevarieties of a ne intersection.

We give examples of proper normal surfaces that are not embeddable into simpli-cial toric prevarieties nor toric prevarieties of aane intersection.