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This smooth projective plane curve has no real rational points: a sum of three real squares is zero only when all three coordinates vanish, which is forbidden in projective space. It is therefore not isomorphic over $\mathbb{R}$ to the projective line, which has rational points. Over $\mathbb{C}$ it becomes a nonsingular conic isomorphic to a projective line. Thus the ground field changes whether a geometric genus-zero curve admits a projective-line parametrization.

In dimension $d$, the fan whose rays are generated by $e_1,\dots ,e_d,-{\sum }_i{e}_i$ and whose cones are generated by proper subsets of those rays is complete. It is the fan of projective space ${\mathbb{P}}^d$.

A smooth projective surface is a two-dimensional smooth variety admitting a closed embedding in projective space. Its integral curves are Cartier divisors, and their intersections, arithmetic adjunction and point blowups control its birational geometry.

A line bundle is very ample when its global sections define a closed embedding into projective space.