I'm working on F. Kirwan's Complex Algebraic Curves. To define intersection multiplicity, Kirwan choose some special projective coordinates and calculate the resultant. She claims before the definition that "In order to show that the definition is independent of the choice of coordinates we show that it is uniquely determined by the properties listed in the following theorem". That's Theorem 3.18.
However, I can't see why this definition is independent of the choice of coordinates. I can't even agree with the proof for Theorem 3.18. For example, in the existence part of the proof, she tries to prove $I_p(C,D)=I_p(D,C)$ using $\mathcal R_{P,Q}(y,z)=\pm\mathcal R_{Q,P}(y,z)$. However, if I use different projective coordinates to define $I_p(C,D)$ and $I_p(D,C)$, $\mathcal R_{P,Q}(y,z)$ and $\mathcal R_{Q,P}(y,z)$ are not simply determinants with some rows interchanged. So where am I wrong and how should I understand this independence of choice of coordinates?
P.S. I have to presume that you have this book because the proof for Theorem 3.18 is too long to type.
