duality by allowing a more general framing, in principle just given by an effective torus invariant Weil divisor (see Definitions \[def:ftv\] and \[def:wftv\] of framed (ftv) and weak framed (wftv) toric varieties, respectively). In general, such a deformed correspondence, here called *framed* duality ($f$-duality), between framed and weak framed toric varieties, is not involutive, but imposing some further conditions on the framing gives back an involutive duality, here called a *calibrated $f$-process*, incorporating the classical duality between Fano toric varieties as a very particular case.

Consequently, $f$-duality is able to describing an enormous number of $f$-mirror symmetric pairs of not necessarily Calabi-Yau hypersurfaces and complete intersections in toric varieties, sensibly improving the current knowledge of mirror partners of non [Calabi-Yau ]{}varieties (see e.g. [@Givental96], [@Hori-Vafa], [@Seidel], [@Efimov], [@KKOY], [@Krawitz], [@HSSW], [@GKR])

For instance, as a very particular but interesting case, a generic projective hypersurface in $\P^n$ of degree $d\geq n+1$, can be thought of a framing of $\P^n$, so admitting (at least) one $f$-mirror dual partner given by an hypersurface in a suitable finite quotient of a weighted projective $n$-space, whose weights are essentially assigned by the framing itself (see §\[sez:ipersuperfici\]). This construction turns out to nicely extending, the pivotal 