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

In this paper, we study tropicalisations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximaldimensional geometric type, show that they can generically have only finitely many singular points, and describe all possible locations of singular points. More precisely, we show that singular points must be either vertices, or generalized midpoints and...

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric l.c.i.-singularities. Our proof makes use of Nakajim...

We compute the first and second homotopy groups of a class of contact toric manifolds in terms of the images of the associated moment map.

Let (P∆, ω) be a toric variety whose moment map image (with respect to the toric Kähler form ω) is the real convex polyhedron ∆ ⊂ MR. Also assume that the anti-canonical class of P∆ is represented by an integral reflexive convex polyhedron ∆0 ⊂ M and the unique interior point of ∆0 is the origin of M . Integral points m ∈ ∆0 correspond to holomorphic toric sections sm of the anticanonical bundl...

We extend the theory and the algorithms of Border Bases to systems of Laurent polynomial equations, defining “toric” roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize tori...

Speyer and Sturmfels [SpSt] associated Gröbner toric degenerations Gr2(C) of Gr2(Cn) to each trivalent tree T with n leaves. These degenerations induce toric degenerations M r of Mr, the space of n ordered, weighted (by r) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe ...

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric variety near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi-Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of th...