Take the 2-minute tour ×  My undergraduate number theory class
decided to dip into a bit of algebraic geometry to finish up the
semester. I'm having trouble understanding this bit of
information that the instructor presented in his notes.  Here it
is in paraphrase (assume we are over an abstract field k)  We
take a polynomial in k, $f =Y^2 - X^3 -aX -b$ and homogenize the
polynomial to $F = Y^2Z - X^3 -aXZ^2 - bZ^3$. Note that the
points at infinity of V(F) consist of triples $[\alpha : \beta :
0]$ s.t $ -\alpha^3 = 0$, hence the only point at infinity is
$[0 : 1 :0]$  The part I'm confused about is in italics. He
introduces the terms "points at infinity" without defining it.
After some google time, I understand what a point at infinity
means in the context of a projective space/projective line but
am having trouble understanding how the professor came to his
conclusion about the point at infinity in this particular
example  Here is my question. In general, are all points in the
locus of vanishing points for