The equation $y^2 = x^3 + 1$ defines an elliptic curve over $\mathbb{C}$. I could define a line bundle by considering polynomials of "degree" 3 on this space. How could I do this more precisely? I have defined an equation in two variables, cutting out a section in affine space:
$$ \{ (x,y) : y^2 - (x^3 + 1) = 0 \} \subseteq \mathbb{C}^2 $$
and in the big space I know that I could use any polynomial in two variables, $\mathbb{C}[x,y]$ which is spanned by $x^m y^n$ and there should be a notion of degree:
$$ \deg (x^m y^n ) = m + n $$
and this could correspond to the a $\mathcal{O}(-k)$ with $k = m+n$. So now we need the pullback of that sheaf. Not knowing any better I'm going to say that:
$$ \deg y^2 = \deg x^3 = 3 $$
Naively I could try to write line bundles of various ranks and hope for the best:
$\langle 1 \rangle$
$\langle x \rangle$
$\langle x^2 \rangle$
$\langle x^3, y^2 \rangle $
$\langle x^4 \rangle$
but now I have to say something like $\deg (x^3 + 1) = 3$ which still seems correct but troublesome. What about if I arrange them into a sort of ascending chain of ideals:
- $\{ 0\} \subseteq \langle 1 \rangle \subseteq \langle 1,x \rangle \subseteq \langle 1,x,x^2 \rangle \subseteq \langle 1,x,x^2,x^3, y^2 \rangle \subseteq \langle 1,x,x^2, x^3, y^2,x^4, xy^2 \rangle \subseteq \mathbb{C}[x,y]$
Gröbner bases have only been around since 1965, but I'm not trying to do anything too compicated here.