The canonical strip, I

We introduce a canonical strip hypothesis for Fano varieties. We show that the canonical strip hypothesis for a Fano variety implies that the zeros of the Hilbert polynomial of embedded Calabi--Yau and general type hypersurfaces are located on a vertical line. This extends, in particular, Villegas's `polynomial RH' for intersections in projective spaces to the case of CY and general type hyperplane sections in Grassmannians. We state a few conjectures on the Ehrhart polynomials of certain fan polytopes.