Invariance of the local intersection multiplicity

For (a): multiplying $F$ by $\lambda \in k^{\times }$ multiplies the local equation $f$ by the unit $\lambda$ of $O$, and similarly for $G$; the ideal $(f,g)$ is therefore unchanged, and $I_p={\mathrm{\ell }}_O(O\mathrm{/}(f,g))$ is unchanged.

For (b): if $f^{\mathrm{\prime }},g^{\mathrm{\prime }}$ generate the same ideal as $f,g$, then the ideals $(f,g)$ and $(f^{\mathrm{\prime }},g^{\mathrm{\prime }})$ are equal, so the quotients $O\mathrm{/}(f,g)$ and $O\mathrm{/}(f^{\mathrm{\prime }},g^{\mathrm{\prime }})$ are equal rings and have the same length. In particular $f+ag$ with $g$ generates the same ideal because $f=(f+ag)-ag$, and multiplying $f$ or $g$ by a unit does not change the ideal.

For (c): let $O^{\mathrm{\prime }}$ be the local ring computed in another chart containing $p$, or after an affine change of coordinates at $p$. The chart transition and coordinate changes induce ring isomorphisms $O\to O^{\mathrm{\prime }}$ carrying the local equations of $C$ and $D$ to local equations, hence carrying the ideal $(f,g)$ to the corresponding ideal $(f^{\mathrm{\prime }},g^{\mathrm{\prime }})$; length is invariant under ring isomorphism, so the two computations agree.

For (d): a projective change of coordinates $A$ induces an isomorphism of the local ring at $p$ with the local ring at $A(p)$ and carries local equations of $C,D$ at $p$ to local equations of $A(C),A(D)$ at $A(p)$ [F3]; since length is invariant under isomorphism, $I_p(C,D)=I_{A(p)}(A(C),A(D))$, with finiteness preserved on both sides by [F4].

Statements (a)–(d) are proved in steps 1.1, 1.2, 2.1 and 2.2, so $I_p(C,D)$ depends only on the curves and the point, not on the chosen defining forms, local equations, chart, affine coordinates or projective coordinates.