Fano Variety - The Example of Projective Hypersurfaces

The Example of Projective Hypersurfaces

The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of is, which is very ample (its curvature is n+1 times the Fubini–Study symplectic form).

Let D be a smooth Weil divisor in, from the adjunction formula, we infer, where H is the class of the hyperplane. The hypersurface D is therefore Fano if and only if .