Tangent lines of a smooth curve $C \subseteq \mathbb{P}^2$

Question

Let $C$ be a smooth curve, given as the zero locus of a homogeneous polynomial $f \in \mathbb{K}[x_0,x_1,x_2]$.

Consider the morphism $\varphi_C:C\rightarrow\mathbb{P}^2$ such that $P\mapsto(\frac{\partial f}{\partial x_0}(P):\frac{\partial f}{\partial x_1}(P):\frac{\partial f}{\partial x_2}(P))$.

Find a geometric description of $\varphi_C$.

What does it mean geometrically if $\varphi_C(P)=\varphi_C(Q)$ for two distinct points $P,Q \in C$?

I have an idea for the answer.

Consider the line $l_{C,P}$ of equation $\frac{\partial f}{\partial x_0}(P)x_0+\frac{\partial f}{\partial x_0}(P)x_1+\frac{\partial f}{\partial x_0}(P)x_2=0$, which is the tangent line of $C$ on the point $P$.

So, $\varphi_C$ encodes al the tangent lines of $C$, and in particular $\varphi_C(P)$ gives the coefficients of the tangent line on $P$.

Is this the correct answer?

For the second part, it is true that $P$ and $Q$ have the same tangent line, but can we say more than this?

Thanks!

Answer 1

The morphism $\varphi_C$ is best seen as a morphism $\varphi_C:C\rightarrow(\mathbb{P}^2)^*$ from the curve $C\subset \mathbb P^2$ to the dual projective space $(\mathbb{P}^2)^*$, whose typical point $Q=(a:b:c)\in(\mathbb{P}^2)^*$ corresponds to the line $l_Q\subset \mathbb P^2$ of equation $ax+by+cz=0$.
If $C$ is a line $ax+by+cz=0$, then the morphism $\varphi_C$ has constant value $\varphi_C(P)=(a:b:c)$ for all $P\in C$.
But in all other cases (if char. $\mathbb K=0$ ) the image of $\varphi_C$ is a curve $C^*\subset (\mathbb{P}^2)^*$, called the dual curve to $C$.
If $C$ has degree $d\gt1$, then $C^*$ has degree $d(d-1)$, an easy consequence of Bézout's theorem.
If you recall that the degree of $C^*$ is its number of points of intersection with a line in $(\mathbb{P}^2)^*$ you obtain the intersting result that there are $n(n-1) $ tangents to $C$ passing through a given general point of $\mathbb P^2$: this number is classically known as the class of $C$.

Finally, let me remark that in order to have a perfect duality between $C$ and $ C^*$ one has to define $\varphi_C$ also for singular $C$ and one then obtains beautifully symmetric formulas relating the invariants of $C $ to those of $C^*$: the Plücker formulas.
Dual curves are explained in elementary books on classical algebraic geometry.
Two excellent ones are Fischer's Plane Algebraic Curves and Brieskorn-Knörrer's identically named text.
Wikipedia also has a nice article on Plücker's formulas

Edit
The number of double tangents to the curve $C$ is the number of ordinary nodes of $C^*$ and that number is, according to the Plücker formulas, $\frac 12 d(d-2)(d-3)(d+3)$.
This number is zero only if $C$ is a conic or a cubic.