TITLE: Projective closure of affine curve
QUESTION [6 upvotes]: Is there a generalized method to find the projective closure of an affine curve? For example, I read that the projective closure of $y^2 = x^3−x+1$ in $\mathbb{P}^2$ is $y^2z = x^3−xz^2+z^3$.
If I want to find the the closure of another affine curve, what method should I employ? I can't seem to find an adequate description in any text.
Thanks

REPLY [4 votes]: Let me add a comment with some geometric flavor to Charles's clean answer. Once the field is algebraic closed, then the affine curve $C\subset \mathbb{A}^2$ is never compact. However it admits an embedding into the projective space (remember that we know that $\mathbb{A}^2\subset \mathbb{P}^2$) where we can look for a compactification. The construction above, geometrically speaking, takes a hyperplane in $\mathbb{P}^2$ and compactifies with it. In other words, there is a hyperplane (divisor) $H=[z=0]\subset \mathbb{P}^2$ which tells you which points to add to your curve $C$, in order to get it compact: let's denote it by $\tilde{C}$. At the end of the day, you will have that $\tilde{C}\cap H$ are the points that you have added to $C$.
In a more general setting (related to the Groebner basis), if you have a manifold $M$, you may ask yourself that it would be desirable for a compactification of it to be algebraic. Now, as far as I now, there are two main obstructions for a compactification to be algebraic. There may not be a "nice" compactification and it may not have enough meromorphic functions. By "nice" here, I mean a compactification given by divisors (with normal crossings usually is asked), and by "enough meromorphic functions", there may not be ample line bundles. Questions on -When such obstructions do not cause troubles?- are big theorems in algebraic geometry. Not referring to self-confidence issues... "here, positivity helps a lot!"