Projective completion

A projective completion of an affine variety $X$ is a projective variety containing $X$ as a dense open subvariety. Homogenizing affine defining equations and taking their projective zero set gives the projective closure, although it may add singularities at infinity.

The homogenization of a polynomial $f(x_1,\dots ,x_n)$ of total degree $d$ is the homogeneous polynomialIts projective zero set is the projective closure of the affine hypersurface defined by $f$.