Hypersurfaces in projective space

A degree d hypersurface in projective space is the subvariety of Pn of a (usually irreducible) homogeneous polynomial of degree d.

Moduli
The moduli space of degree d hypersurfaces in Pn is isomorphic to the $$\left(\binom{n+d+1}{d}-1\right)$$-dimensional projective space of degree d homogenous polynomials in n+1 variables, up to scaling.

Special cases

 * Plane curves
 * Surfaces in P3
 * Hypersurfaces in P4
 * Hyperplanes in projective space
 * Quadric hypersurfaces
 * Cubic hypersurfaces
 * Quartic hypersurfaces
 * Quintic hypersurfaces

Generalizations

 * Subvarieties of projective space