I'm going through Knapp's book on elliptic curves and I got stuck in a minor detail.

This is a part of the proof of Proposition 2.12:

I could understand everything except for this little detail: Where are we making use of the assumption $d>2$?

I will post some pictures about the references that the proof makes use of, in order for you to understand the whole argument.

Proposition 2.7 and identity (2.12):

Lemma 2.11:

enter image description here

  • $\begingroup$ Well, a curve of degree $2$ doesn’t even have any flexes. $\endgroup$ – Lubin May 25 at 15:30
  • $\begingroup$ Yes, but that is a proposition that Knapp proves after this one, so the argument to understand Prop. 2.12 should omit that assertion. Moreover, here "curve" does not mean that the corresponding polynomial is necessarily irreducible, at least at this section of the book. $\endgroup$ – solomeo paredes May 25 at 15:41

If $d = 1$ then the Hessian is the zero matrix at every point, so Lemma 2.11 does not apply. And the case $d = 2$ is discussed immediately after the end of your first image. In this case $F$ itself defines a conic and the author concludes that every nonsingular point of $F = 0$ is a flex.

  • $\begingroup$ I already know that the case $d=2$ is handled immediately after what I posted. That is not what I was asking. My question was not about the case $d=2$. I was asking how the assumption $d>2$ is used in the argument I posted. $\endgroup$ – solomeo paredes May 26 at 14:02

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