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I am trying to do Exercise 5.33 on page 64 of Fulton's book on algebraic curves.

5.33 Let $C$ be an irreducible cubic, $L$ a line such that $L\bullet C = P_1+ P_2 + P_3,$ $P_i$ distinct. Let $L_i$ be the tangent line to $C$ at $P_i:$ $L_i \bullet C = 2P_i + Q_i$ for some $Q_i.$ Show that $Q_1, Q_2, Q_3$ lie on a line. ($L^2$ is a conic!)

I would appreciate any help in solving this problem.

Some background information for convenience: We recall that for two projective plane curves of degree with no common component, we have defined $F\bullet G$ to be the formal sum

$$F\bullet G = \sum_{P\in \mathbb{P}^2} I(P,F\cap G) P. $$

where $I(P, F\cap G)$ is the intersection number of $F$ and $G$ at $P.$ The intersection number is $1$ if and only if $F$ and $G$ meet at $P$ transversally, which means that $P$ is a simple point on both $F$ and $G,$ and the tangent to $F$ at $P$ is distinct to the tangent to $G$ at $P.$ The first condition in our problem means that $L$ intersects $C$ transversally at $3$ distinct points $P_i.$ The simplest version of this problem is when $P_i\neq Q_i, i=1,2,3$ (i.e when each tangent $L_i$ meets $C$ again at another points $\neq P_i$) It then says: Show that the three points when the tangents again intersect $C$ lie on a line.

I'm not sure how to proceed, many things seem like they could be relevant (Max Noether's theorem has just been covered and Bezouts right before that), especially this one:

Proposition 2. Let $C, C'$ be cubics, $C'\bullet C = \sum_{i=1}^9 P_i;$ suppose $Q$ is a conic, and $Q\bullet C = \sum_{i=1}^6 P_i.$ Assume $P_1,\cdots, P_6$ are simple points on $C.$ Then $P_7, P_8, P_9$ lie on a straight line.

Can someone please help me with this question?

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  • $\begingroup$ Let $C'=L_1L_2L_3$, and let $Q=L^2$. Then just apply Proposition 2. Note that the $P_i$ are simple points on $C$ (by Property (5) of the intersection number, p.37). $\endgroup$ – Nils Matthes Jun 3 '13 at 12:48
  • $\begingroup$ @NilsMatthes It was right in front of me! Thanks Nils. $\endgroup$ – Katie Dobbs Jun 3 '13 at 13:13
  • $\begingroup$ You're welcome. $\endgroup$ – Nils Matthes Jun 3 '13 at 13:18
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Yes, apply your Proposition with $C'=L_1L_2L_3$ and $Q=L^2$.

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  • $\begingroup$ Ahh, the answer was easier than I expected. Thanks Ted! $\endgroup$ – Katie Dobbs Jun 3 '13 at 13:14

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