Projective spaces

Presentations

 * Projective space Pn is given by the Proj construction applied to the ring C[x0,x1,...,xn].
 * Projective space is the quotient of Cn+1\{0} by C*.

Closed subvarieties

 * Hypersurfaces in projective space
 * Complete intersections in projective space

Special cases

 * P1
 * P2
 * P3
 * P4

Generalizations

 * Weighted projective spaces
 * Toric varieties
 * Grassmannians
 * Projective varieties
 * Spaces with a torus action
 * Spaces with an affine stratification