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I want to find the inflection points(i.e. $I_P(C,T_P C) \ge 3$) of cubic curve $x^3+y^3+z^3=0$ in $\mathbb{P}^2_k$

If $(char(k),2)=1$, I know that the smooth inflection points are equal to the intersection points with Hessian curve. Since the curve is smooth, we can find exactly all inflection points.

But how can I find them if $char(k)=2$?

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    @BabakSorouh Yes, I edited it.2012-12-01
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    You mean if $\text{char}(k) = 3$, right?2012-12-01
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    @MichaelJoyce No, I can find them in the case of $char(k)=3$ by computing some Hessian. But if $char(k)=2$, then I can't use Hessian and the only method I can do is using definition and direct computation, which seems hard.2012-12-01
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    Okay, but note that in $\text{char}(k) = 3$, your curve is simply the line $x + y + z = 0$ 'with scheme structure' because $x^3 + y^3 + z^3 = (x + y + z)^3$ when $\text{char}(k) = 3$. So it's really a 'triple line', which is singular at all points.2012-12-01
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    @MichaelJoyce I missed it, thanks!2012-12-02

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