= Projective completion
= Projective closure
{synonym}
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 $Z\ne0$ of $\mathbb P^n$.
If $I$ is the defining <ideal>, homogenize every member of $I$. If instead $f_1,\ldots,f_r$ 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.<Elementary Uses of Gröbner Bases in Macaulay2>[projective-closure and saturation discussion]{ref}
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