Let $X^n \subset \mathbb{P}^N$ to be a toric projective variety. Is $X$ a local complete intersection? Is being a local complete intersection an intrinsic property, independent of embedding?



Are projective toric varieties, locally complete intersection?
No, there are projective toric varieties that are not even Gorenstein (and hence not l.c.i). In fact the Gorenstein property translates into a geometric property of the defining polytope, namely that it is reflexive under taking the dual polytope. 


