hypersurface.

Anyway, this is a case by case checking, quickly becoming essentially impossible for more general hypersurface and complete intersections in toric varieties, due to the wild singularities $f$-duality produces. According with Chiodo and Ruan [@Chiodo-Ruan], it is generally believed that considering suitably associated LG models may sensibly simplify singularities and give rise to alternative way of checking mirror symmetry. The already mentioned LG/Hypersurface correspondence, presented in §\[ssez:K-dualità\], allows one to drawing an alternative conjectural approach to checking mirror symmetry (see Remark \[rem:K-dualità\]). A more detailed and rigorous exposition of these aspects is postponed to next papers.

This paper is organized as follows. §\[sez:preliminari\] is devoted to introduce the needed notation on toric varieties, their divisors, hypersurfaces and associated stratifications. §\[sez:dualita-ftv\] is dedicated to the definition of framed toric varieties and framed duality. In §\[sez:framingPn\] an important class of framed toric varieties admitting a calibrated $f$-process is presented, namely projective spaces endowed with suitable framings. Then §\[sez:dualita-hyp\] is devoted to present mirror symmetric consequences of $f$-duality for hypersurfaces in complete toric varieties. In the following §\[sez:ipersuperfici\], all these considerations are applied to the important class of examples given by hypersurfaces in $\P^n$ of degree $d\ge n+1$. Then in §\[sez:CI\], $f$-duality 