Question: (Exercise 3.3.16, p.139, of Algebraic Geometry: A Problem-Solving Approach by Garrity et al) Show that for any polynomials $f$ and $g$ in $\mathbb{C}[x,y]$ and a point $p$ in $\mathbb{C}^2$, for any affine change of coordinates $T$ we have $$I(p, V(f) \cap V(g)) = I(T(p), V(T^{-1} f) \cap V(T^{-1} g))$$ Hint: this problem is actually not that hard. Its solution involves little or no calculations.
I fail to see at all how one can prove or disprove this using the axiomatic definition of intersection multiplicity at all, since the definition only refers to a single point, but this problem asks to compare calculations at different points.
Is the idea simply supposed to be that for all of the axioms, an affine change of coordinates leaves their corresponding conditions invariant, i.e. $p$ lies on a common component of $V(f)$ and $V(g)$ if and only if $T(p)$ lies on a common component of $V(T^{-1}f)$ and $V(T^{-1}g)$?
How can we show this invariance for each of the axioms using the definition of affine transformation and change of coordinates? (i.e. using formulas?)
It says briefly (on p. 60) of Kirwan's Complex Algebraic Curves, that
Since the conditions are independent of the choice of coordinates we may assume that $p = [0\ 0\ 1]$.
It also says that we can assume that $f$ and $g$ are irreducible by 3. and 4. (using the number of axioms I give below), that $I(p, V(f) \cap V(g))$ is finite by 1., and that $I(p, V(f) \cap V(g)=k>0$ by 2., and that by induction on $k$ we may assume that any intersection multiplicity strictly less than $k$ can be calculated using only the conditions 1.-6. (The idea being to show that the intersection multiplicity can always be calculated using only these axioms, and thus that these axioms must determine the intersection multiplicity completely.)
Just like in Garrity et al, no careful proof is given of any of these claims. I get the "idea" behind all of them, none of them seem shocking and they all seem plausible, but I don't think that I know how to prove them rigorously if I needed to do so.
Context: This might be easier using the definition of intersection multiplicity in terms of resultants -- this is not what I am asking about here.
I am talking about the axiomatic definition (which turns out to be equivalent to that given in terms of resultants). I am copy-pasting the definition given in Garrity et al, Algebraic Geometry: A Problem Solving Approach p. 138, theorem 3.3.12, although similar definitions can be found on Wikipedia and in Kirwan's Complex Algebraic Curves, Theorem 3.18, p. 59:
Given polynomials $f$ and $g$ in $\mathbb{C}[x,y]$ and a point $p$ in $\mathbb{C}^2$, the intersection multiplicity is the uniquely defined number $I(p, V(f) \cap V(g))$ such that the following axioms are satisfied:
1. $I(p, V(f) \cap V(g)) \in \mathbb{Z}_{\ge 0}$, unless $p$ lies on a common component of $V(f)$ and $V(g)$, in which case $I(p, V(f) \cap V(g) = \infty$.
2. $I(p, V(f) \cap V(g)) = 0$ if and only if $p \not\in V(f) \cap V(g)$.
3. Two distinct lines meet with intersection multiplicity one at their common point of intersection.
4. $I(p, V(f) \cap V(g)) = I(p, V(g) \cap V(f))$.
5. $I(p, V(f) \cap V(g)) = \sum r_i s_j I(p, V(f_i) \cap V(g_j))$ when $f= \prod f_i^{r_i}$ and $g = \prod g_j^{s_j}$.
6. $I(p, V(f) \cap V(g)) = I(p, V(f) \cap V(g+af))$ for all $a \in \mathbb{C}[x,y]$.
Answer
I struggled with this too. It seems like an elaborate induction could be done. First note that Axioms 1, 2, 3 and 4 do not depend on a choice of coordinates. The basis step of the argument would be to invoke Axiom 3 to show that $I(p,C,D) = I(Tp,TC,TD)$ when $C$ and $D$ are lines. If the curves $C$ and $D$ had larger degree than one then use Axioms 5 and 6 to reduce the computation of $I(Tp,TC,TD)$ to that of smaller degree curves and invoke the induction hypothesis.
Also, I believe that Kirwan's proof of uniqueness could avoid choosing a favored coordinate system, although it would notationally more burdensome. With that, one could show that for a given transformation the function $I(T\cdot,T\cdot,T\cdot)$ satisfys the same axioms as $I$ and so they have to be the same function.
Either way, I don't think either of these things is what the authors had in mind.
Finally, Fulton includes invariance under change of coordinates as an axiom. This seems like a reasonable thing to demand, and given that the point of writing these axioms is not to get away with the fewest number of axioms, but to sweep the details of a good definition under the rug, this seems like a reasonable thing to throw into the mix. Indeed, given a good definition, $I(p,V(f),V(g)) = \dim_k k[x,y]_p/(f,g)$, it is clear that the ring on right will be isomorphic after applying an affine change of variables.
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