Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

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Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

Mirror Symmetry for Two Parameter Models - II *

The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the chamber decomposition of the cone of divisors in Mori theory is naturally identified with the decomposition of the G-ample cone from geometric invariant theory. In particular every rational contraction of a Mori dream space comes from GIT, and all possible factorizations of a rational contraction can be read off from the chamber decomposition.

https://projecteuclid.org/download/pdf_1/euclid.mmj/1030132722

Mori dream spaces and GIT.

Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties

/pdf/linear-gale-transforms-and-gelfand-kapranov-zelevinskij-5deoa2f4if.pdf

Linear gale transforms and gelfand-kapranov-zelevinskij decompositions

This paper was submitted to the Oberwolfach Conference "Combinatorial Convexity and Algebraic Geometry", October 1997. 
Let $M={\mathbb Z}^r$. For convex lattice polytopes $P,P'$ in ${\mathbb R}^r$, when is $(M \cap P)+ (M \cap P') = M \cap (P + P')$? 
Without any additional condition, the equality obviously does not hold. 
When the pair $(M,P)$ corresponds to a complex projective toric variety $X$ and an ample divisor $D$ on $X$, it is reasonable to assume that $P'$ corresponds to an ample (or, more generally, a nef) divisor $D'$ on the same $X$. Then the question correspons to the surjectivity of the canonical map \[ H^0(X,{\mathcal O}_X(D))\otimes H^0(X,{\mathcal O}_X(D'))\to H^0(X,{\mathcal O}_X(D+D')).\] 
When $X$ is nonsingular, the map is hoped to be surjective, but this remains to be an open question after more than ten years. 
The paper explores various variations on the question in terms of toric geometry.

Problems on Minkowski sums of convex lattice polytopes

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=19&iss=3&rank=12

Hironaka's Additive Group Scheme, II

/pdf/torus-embeddings-and-tangent-complexes-40cl0sjjw4.pdf

Tadao Oda

Papers

Linear gale transforms and gelfand-kapranov-zelevinskij decompositions

Problems on Minkowski sums of convex lattice polytopes

Hironaka's Additive Group Scheme, II

Torus embeddings and tangent complexes

Simple convex polytopes and the strong Lefschetz theorem