Greene-Plesser description of a mirror partner of the quintic threefold [@GP], to higher degrees. Moreover, for lower degrees, a generic projective hypersurface in $\P^n$ of degree $d\leq n$ can be thought of a weak framing of $\P^n$, whose associated $f$-mirror dual partner can no more be a complete variety (see §\[ssez:NegKod\]), sharing strict relations with Landau-Ginzburg (LG) mirror models proposed by Givental [@Givental96], [@Givental-ICM].

Furthermore, $f$-duality turns out to extending many known dualities between Calabi-Yau hypersurfaces and complete intersections in toric varieties: this is the case of the Berglund-Hübsch duality [@Berglund-Hubsch] and, more generally, of the recent Artebani-Comparin-Guilbot duality [@ACG] (see §\[ssez:BHK\] and \[ssez:ACG\])).

More in general, $f$-duality opens up to a lot of stimulating connections with many expects of the current status of art of research in mirror symmetry. For instance, it suggests a suitable re-parameterizations of the Landau-Ginzburg mirror model of a complete intersection in a toric variety, proposed by Hori and Vafa in 2000 [@Hori-Vafa], in such a way that, at least for projective hypersurfaces, $f$-duality turns out to exhibit a suitable compactification of this re-parameterized LG mirror model, extending to higher degrees what observed by Hori and Vafa for [Calabi-Yau ]{}projective hypersurfaces (see §\[ssez:HoriVafa\]). Since mostly 