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A projective completion of an affine variety $X$ contains $X$ as a dense open subvariety of a projective variety. For a chosen embedding $X\subseteq {\mathbb{A}}^n$, its projective closure is the closure inside the standard chart $Z\ne 0$ of ${\mathbb{P}}^n$.

If $I$ is the defining ideal, homogenize every member of $I$. If instead $f_1,\dots ,f_r$ is a chosen generating list, the homogenized generators can leave extra components at infinity. Saturating their generated ideal by $Z$ removes those components.

For example, $(x,xy-y)=(x,y)$ defines the origin in affine space ${\mathbb{A}}^2$. Homogenizing only the displayed generators gives $(X,XY-YZ)$. Its equations in the projective plane have both $[0:0:1]$ and the extraneous point $[0:1:0]$. Saturation by $Z$ gives $(X,Y)$, the correct closure. Even a correct projective closure can acquire singularities at infinity.^[ref]