Why do we ask $w\nmid F$ in order for $i\left (\left (0:1:0\right ),w,F\right )$ to be defined?

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?

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