is extended to complete intersections subvarieties in complete toric varieties. Finally in §\[sez:open\] many further considerations and open problems are collected: in particular in §\[ssez:NegKod\] weak framed toric varieties are defined, with an application to hypersurfaces in $\P^n$ of degree $d\le n$ and connections with Givental’s LG mirror models.

It is a pleasure to thank M. Artebani for several clarifications about many aspects treated in [@ACG] and T. Hübsch for his interested comments and interesting suggestions, giving rise to perspectives in §\[ssez:Hubsch\]. Many thanks also to S. Filippini, for useful conversation during her last visit in Turin, and to G. Bini for his considerations. I am also indebt with D. Grinberg and M. Riedel, via the `Math Stack Exchange` platform, for useful hints in proving combinatorial Lemmas in §\[ssez:combinatorica\]. Many computations and proofs’ prototypes have been partially performed by means of several Maple routines, mostly of them jointly written with L. Terracini, and some of them based on the Maple package `Convex` [@Convex].

Preliminaries and notation on toric varieties {#sez:preliminari}
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A *$n$–dimensional toric variety* is an algebraic normal variety $X$ containing the *torus* $T:=(\C^*)^n$ as a Zariski open subset such that the natural multiplicative self–action of the torus can 