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I'm beginning to study algebraic curves and I couldn't prove if we have $L$ a line bitangent to $F$, i.e, there are points $p_1, p_2\in F$, such that $L=T_{P_1}F=T_{P_2}F$, then $P_L\in F^\vee$ is a nodal point.

I'm starting to prove this point is singular. I know the point $P_L$ in the dual space corresponding to the tangent $T_{P_1}F$ (or $T_{P_2}F$) and it's given by $P_L=\bigg(\frac{\partial F}{\partial X}(P_1):\frac{\partial F}{\partial Y}(P_1): \frac{\partial F}{\partial Z}(P_1)\bigg)$, but I don't know how to use the fact $L=T_{P_1}F=T_{P_2}F$ to prove $\frac{\partial F}{\partial X}(P_L)=\frac{\partial F}{\partial Y}(P_L)=\frac{\partial F}{\partial Z}(P_L)=0$.

I really need help.

Thanks

user122342
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1 Answers1

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You don't need any algebra to proof this, it is by definition.

When two tangents in two distinct points (say: $P_1, P_2$) are equal (say: $L$) this means per defintion that in the dual space one point - namely the dual point to $L$ - has two tangents: the dual lines of $P_1, P_2$. A point with two tangents is a node (also per definition).

Gerard
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