I am currently reading Kirwan's Complex Algebraic Curves, to understand the proof of Bézout's theorem given in the expository paper here : https://www.math.ucla.edu/~ggim/F11-214A.pdf .
In the paper, Bézout's theorem is proved for plane projective curves over a generally algebraic closed field $k$; however he references Kirwan's book in his proof, which only develops the theory over complex projective curves. This is fine for the most part, except when the author assumes that projective transformations can be used throughout to change coordinate systems so that certain points are not on the curves, and specific axes are tangent to specific point on the curves.
My question is, since in Kirwan these properties ultimately follow from the quotient topology defined on $\mathbb{P}_n (\mathbb{C})$ by $\mathbb{C}$, do they also transfer to when $k$ is an arbitrary algebraically closed field? I'm aware Bézout's theorem holds for any projective plane curves over an arbitrary field, but I am specifically looking to understand the methods used in Gim's paper.
Answering for completion:
Kirwan's use of projective change of coordinates throughout ultimately follows from the following theorem:
Theorem: Given four distinct points $p_0, p_1, p_2$ and $q$ in $\mathbb{P}^{2}(k)$, no three of which are are collinear when viewed as points in $k^{2+1}$, there exists a projective transformation sending $p_0, p_1, p_2$ to $[1:0:0], [0:1:0],[0:0:1]$, and $q$ to $[1:1:1]\,$ respectively.
A proof for which in the complex case can be found in his book cited above.