Problem 7-5 Algebraic Curves William Fulton

Question

I have been trying problem number 5 of chapter 7 of the Fulton, I attach the statement in the following link; Problem 7-5 Fulton

QUESTION:

Let P be an ordinary multiple point on C, $f^{-1}(P) = \{P_1,...,P_r\}, L_i = Y -\alpha_iX$ the tangent line corresponding to $P_i = (0, \alpha_i)$ Let $G$ be a plane curve with image $g$ in $\Gamma (C) \subset \Gamma (C') $
a) Show that $ord^{C'}_{P_i}(g)\geq m_p(G)$, with iguality if $L_i$ is not tangent to G at P.
b) if $s \leq r$ and $ord^{C'}_{P_i}(g)\geq s $ for each $i = 1,...,r $ show that $ m_p(G) \geq s $ (Hint: How many tangents would G have otherwise?)

MY STEPS

I'm not very sure, but I'm trying to analyze the order of intersection between the tangent lines $L_i$ and the plane curve $C'$ and I also know that $ord^{C'}_{P_i}(g)\geq 1$ also as $m_p(G)$ is the intersection multiplicity between $C$ and $G$ at the multiple point $P$ and given that an ordinary multiple point has at least two distinct tangent lines, I'm not sure but it seems to me that it fulfills that:

$m_P(G)\leq \sum_{i=1}^{r} ord_{{P_i}}^{C' }(g)$

How could I justify this? Also I could only get so far in part a, any clue or idea on how I can continue?

Answer 1

Here's my solution to this problem:

If $P$ does not go through $G$ then $m_p(G)=0$ and we are done. Thus, assume $m_p(G)=m$, where $m>0$. Write $G=G_m+...+G_n,$ where each $G_i$ is a form of degree i. Further, we can write $G_m=\prod_{i=1}^k(\gamma_i X - \beta_iY)^{j_i}$ where the $\sum_i^k j_i=m$. I will only consider the case where $\beta_i's$ are nonzero, however, the problem does go through with slight modifications if you have the opposite scenario. Thus, with the latter assumption, we can further say $G_m=\prod_{i=1}^k(\gamma_i X - Y)^{j_i}$.

Then, $g \in \Gamma(C')$ can be written as $ g(x,xz)= x^m(\prod_{i=1}^k(z-\gamma_i)^{j_i}+x(...))$

In particular, since the order function turns multiplication into sums, we have $$ord_{P_i}^{C'}(g)=m + ord_{P_i}^{C'}(\prod_{i=1}^k(z-\gamma_i)^{j_i}+x(...)) \geq m = m_p(G) $$.

In this last line, equality holds iff $\prod_{i=1}^k(z-\gamma_i)^{j_i}+x(...)$ is a unit in the local ring at $P_i$, which happens iff its value at $P_i$ is nonzero, which, in turn, happens iff $\gamma_i\neq\alpha_i$. This solves part A.

For part b, assume that the assumption holds but that the $m_p(G)=k<s$. Use the same ideas as in part A here and show that you cannot solve for as many $\gamma_i's$ as it would be necessary to guarantee the assumption.