We classify the smooth Fano 4-folds of Picard number two that have a general hypersurface Cox ring.

The anticanonical complex generalizes the Fano polytope from toric geometry and has been used to study Fano varieties with torus action so far. We provide an explicit description of the anticanonical complex for complete intersections in toric varieties defined by non-degenerate systems of Laurent polynomials. As an application, we classify the terminal Fano threefolds that are embedded into a fake weighted projective space via a general system of Laurent polynomials.

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We classify all smooth Calabi–Yau threefolds of Picard number two that have a general hypersurface Cox ring.