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Knapp's book on elliptic curves, on the fourth section of its second chapter, has a confusing argument which I cannot understand.

For a cubic $F\in k\left [x,y,w\right ]$ (homogeneous third-degree polynomial) and a non-singular point $P\in \mathbb{P}_2\left (k\right )$ in its zero-locus, we say that $P$ is a flex if $i\left (P,L,F\right )\geq 3$, $L$ being its tangent line and $i$ being the intersection multiplicity. This is given at page $35$.

Write $F=c_{xxx}x^3+c_{xxy}x^2y+c_{xxw}x^2w+c_{xyw}xyw+\cdots $

At pages $40$ and $41$ it states several conditions, leading to several conclusions involving the coefficientes of $F$. The first condition is that $P=\left (0,1,0\right )$ be in the zero locus, leading to $c_{yyy}=0$. Then we want $P$ to be a non-singular point with $w$ as its tangent line, which leads to $c_{xyy}=0\neq c_{yyw}$. Finally, we want $P$ to be a flex. This is achieved by imposing $c_{xxy}=0$.

But here comes my confusion: the book then says the following:

The condition that $i\left (\left (0,1,0\right ),w,F\right )$ be defined is that $w$ not divide $F$, hece that $c_{xxx}\neq 0$.

Why do we have that $w\nmid F$? If $w\mid F$ then $i\left (P,w,F\right )=\infty\geq 3$ (as Knapp states at page $33$, on Proposition 2.8), so we have our desired flex even if we do not impose $w\nmid F$.

Maybe Knapp wanted to define the notion of flex by $3\leq i\left (P,L,F\right )<\infty$ and then it would make sense, but in that case it is not true that $P$ is a flex if and only if $f_1\mid f_2$ ($f_i$ being the $i$-th homogenous component of the local affine form of $F$), and that is a property that Knapp makes use of later.

Maybe we should just add the further condition $i\left (P,L,F\right )<\infty$ in the list of things we impose on the cubic?

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  • $\begingroup$ Not related to the question but, cool name dude! $\endgroup$ – Laz Jul 4 at 22:14

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