We provide a sufficient condition for a general hypersurface in a Q-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-Hübsch-Krawitz construction in case the ambient is a Q-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi-Yau hypersurfaces which includes both Batyrev and Berglund-Hübsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes ∆ 1 ⊆ ∆ 2 , where ∆ 1 and ∆ * 2 are canonical.{W = 0} ⊂ P(w)/G ←→ {W * = 0} ⊂ P(w * )/G * ,

We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.

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