In light of the holiday, I would like to air a grievance.
I have no good way to recoordinatize a morphism of varieties as I move between coordinate neighborhoods.
Let me explain what I mean with an example. Consider the affine variety $E_Y$ cut out by the equation
$Z + Z^2 - (X^3 + ZX^2) = 0 \mbox{ }\mbox{ }\mbox{ }\mbox{ } (1)$
The projective closure of $E_Y$, which we'll denote by $E,$ is an elliptic curve with identity element $O:=(0,0)$ in $X,Z$-coordinates.
So there is addition morphism $\mu:E\times E \rightarrow E$ and an inverse morphism $-:E \rightarrow E.$ A calulation shows, $-\mu$ is given by
$X(-\mu) = \frac{(z_1 -z_0)^2 - (x_1 -x_0)(x_1z_0 - x_0z_1)}{(x_1 -x_0)^2 + (x_1 - x_0)(z_1 - z_0)} - x_0 - x_1$
$Z(-\mu) = \frac{z_1 - z_0}{x_1 -x_0}\left(\frac{(z_1 -z_0)^2 - (x_1 -x_0)(x_1z_0 - x_0z_1)}{(x_1 -x_0)^2 + (x_1 - x_0)(z_1 - z_0)} - x_0 - x_1\right) + \frac{x_1z_0 - x_0z_1}{x_1 -x_0} $
on $E_Y \times E_Y$ outside of the locus $(x_1 -x_0)^2 + (x_1 - x_0)(z_1 - z_0) = 0.$
I would like to express the morphism $-\mu$ in terms of regular functions on some open neighborhood of $O \times O,$ but my current method of obtaining such an expression is "to move symbols around" in my expression for $-\mu$ using the relation of the curve, $(1),$ until I obtain a regular expression at $(0,0) \times (0,0).$ This is often a huge waste of time and becomes nearly impossible as the equations defining the variety become more complicated.
So I'm wondering if there is a more methodical way to approach this problem? How does one do this in practice?