I am trying to understand for which $n$ does the curve $F = Y - X^n$ has an inflection point at $P = (0,0)$ as in exercise 3.12 from Fulton's algebraic curves. From a geometric perspective I expect it to be all $n \ge 3$ with $n$ being odd. The tangent is $L = Y$ and $\mathfrak{m}_P = (X)$ as in the proof of theorem 1 in chapter 3.2.
For $n=3$: as I understand that would mean $L = Y = X^3 \in \mathfrak{m}^3$, so $ord_P^F(L) = 3$, so P is an ordinary inflection point.
For $n=4$: wouldn't this be here the same logic, i.e., $L \in \mathfrak{m}^4$? But here P is no inflection point, or what do I miss here?
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G.rald
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2“Inflection point” in this context has a different meaning than it does for smooth curves. (The usual definition in terms of sign changes doesn’t obviously make sense over an arbitrary field, for example.) – Qiaochu Yuan Nov 19 '20 at 09:51
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OK, so the answer in this case would be $P$ is a flex for all $n \ge 3$? – G.rald Nov 19 '20 at 13:15
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Yes, that’s right. – Qiaochu Yuan Nov 19 '20 at 17:55